CCNY Math Review Chapters 5 and 6: Trigonometric functions and graphs

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions CCNY Math Review Chapters 5 and 6: Trigonometric functions and graphs Section 6.: Angles and circles 6..: Angle measure 6..: Angles in standard position 6..: Circle arcs and sectors 6..4: Circular motion 6..5: Chapter 6. Quiz Review Section 6.: Right triangle trigonometry 6..: Right triangle trigonometric ratios 6..: Special right triangles 6..: Chapter 6. Quiz Review Section 6.: Trigonometric functions of general angles 6..: Radian measure of general angles 6..: Trig functions of general angles 6..: Angles in Quadrants,,4 6..4: Finding the reference angle 6..5: Trig functions generalize SohCahToa 6..6: Trigonometric identities 6..7: Evaluating trigonometric functions 6..8: How your calculator computes cosines 6..9: Chapter 6. Quiz Review Section 5.: Trigonometric functions and graphs 5..: Graphing sin θ as a function of angle θ 5..: Graphing y = sin x 5..: Graphing y = A sin x 5..4: Graphing y = sin x 5..5: Four basic sine and cosine graphs 5..6: Transforming trig equations and graphs 5..7: Graphing y = Asin(Bx + C) 5..8: Graphing the standard wave 5..9: Chapter 5. Quiz Review Section 5.5: Inverse trigonometric functions 5.5.: Solving simple trigonometric equations 5.5.: Inverse trigonometric functions 5.5.: Chapter 5.5 Quiz Review 5.5.4: Can this be discarded? All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter V: Trigonometric functions //6 Frame

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 6.. Angle Measure (with Tamara Kucherenko) In geometry, an angle consists of two rays emanating (traveling away) from a common point. In precalculus, an angle is better thought of as the path of a point that moves counterclockwise on a radius circle. The measure of the angle is the length of the path, which could be any positive (or zero) real number. The circumference of a radius circle is, and so the length of the path that goes around the whole circle is radians. half the circle is radians one quarter of the circle is / radians. It s a bit clumsy to keep using the word radian. It is typically written as rad. If an angle measure is written as a pure number, the angle is being measured in radians. O rad O rad O rad Each angle s black initial ray and red terminal ray are drawn as arrows. From the middle picture, rad = 8. The path of a point moving clockwise is also an angle, but its measure is minus the length of the path. How to convert between degrees and radians. 8 = rad rad = ( ) 8 = 8 rad To convert degrees to radians, multiply by 8. To convert radians to degrees, multiply by 8. = 8 = 9 = 6 4 = 45 6 = Example : Express a) 75 in radians; b) /5 radians in degrees. a) 75 = 75 8 = 5 b) 5 = 5 8 = Memorizing the examples listed in the last bullet helps avoid fraction calculations in many cases: Example : Convert a) rad to degrees; b) 4 to radians. a) 4 = 4 = 45 = 495 b) = = 6 = 6 All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6.: Angle Measure 8/5/6 Frame

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 6.. Angles in Standard Position Definition of coterminal angles An angle is in standard position if it is drawn in the xy-plane with its vertex at the origin and its initial side on the positive x-axis. Two angles in standard position are coterminal if their terminal sides coincide. O y x The last two angles above are coterminal. How to check if angles are coterminal O y Angles are coterminal if their measures differ by n radians, or by 6n degrees, where n is an integer. The reason is simple. If you start at the terminal ray of an angle and go either clockwise or counterclockwise one or more circles, you will arrive at the same terminal ray. x O y x Example. Are angles 5 and 7 coterminal? Solution. We look at the difference of the two angles: 5 7 = = 4 =. Yes, the angles are coterminal. To find angles that are coterminal with a given angle, add or subtract a whole number times 6 degrees, or a whole number times rad. Example. Find an angle with measure between and 6 that is coterminal with the angle of measure 9 in standard position. Solution. Keep subtracting 6 from 9 until the resulting angle measure is in the requested range. 9 6 6 6 =. Mathematical language: Example, somewhat clumsily, referred to an angle of measure 9. From now on, we will usually omit the words of measure. All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6.: Angle Measure 8/5/6 Frame

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 6.. Measuring circle arcs and sectors Definition of circular arc In the picture below, the red path is called a circular arc with central angle θ. O θ r s How to find the length of a circular arc Assume θ >. Then an angle with radian measure θ traces a path of length θ on a radius circle. Doubling the radius to doubles the length of the path to θ. In general, if the radius is r, then the path length multiplies by r as well. In a circle of radius r, the length s of an arc with central angle θ (radians) is s = rθ Example : Find the length of an arc of a circle with radius meters and central angle. Solution. To begin, convert into radians: 8 = 6 rad. The length of the arc is s = rθ = () 6 = 5 meters. Definition of circular sector In the picture below, the shaded region is called a circular sector with central angle θ. O θ r s The area of a circle of radius r is r. The fraction of the circle s area that is the area of the sector is the same as the fraction of the circle s circumference that is the length of the arc. Therefore Sector area = θ (area of circle) = θ (r ) = r θ How to find the area of a circular sector In circle of radius r, the area A of a sector with central angle θ (radians) is A = r θ Example 4: Find the area of a sector of a circle with central angle 6 and radius meters. Solution. To begin, convert 6 into radians: 6 8 = rad. So the area of the arc is A = r θ = ( ) () = square meters. All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6.: Angle Measure 8/5/6 Frame 4

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 6..4 Circular Motion O θ s r There are two ways to describe motion of a point moving at constant speed along a circle: Linear speed = distance traveled along the circle time. Angular speed = change of angle (in radians) time. How to find linear speed and angular speed Suppose a point moves along a circle of radius r and moves θ radians in time t. Let s = rθ be the distance the point travels in time t. Then the object s Angular speed ω = θ t Linear speed v = s t Example 5: A boy rotates a stone in a -ft-long sling at the rate of 5 revolutions every seconds. Find the angular and linear velocities of the stone. Solution. In seconds, the angle θ changes by 5 = radians. Angular speed = ω = θ t = rad = rad/second. seconds The distance traveled by the stone in seconds is s = 5 r = 5 = 9 feet. Linear speed= v = s t = 9 feet = 9 feet/second. seconds How to convert angular speed to linear speed If a point moves along a circle of radius r with angular speed ω, then its linear speed v is given by v = rω. Example 6: A bicycle s wheels are inches in diameter. If the wheels rotate at 5 revolutions per minute find the speed of the bicycle in miles per hour. Solution. Each revolution is radians, and so the wheels angular speed is 5 = 5 rad/min. Since the wheels have radius inches, the linear speed is v = rω = 5 inches/minute. To convert inches/minute to miles/hour recall that mile = 58 feet = 58 = 8 6 inches, while hour = 6 minutes. Thus 5 inches minute = 5 inches mile minute 58 inches 6 minutes hour = 5 6 8 miles = 5 6 hour 48 miles = 5 miles hour 48 hour All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6.: Angle Measure 8/5/6 Frame 5

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 6.. Right triangle trigonometric ratios (with Tamara Kucherenko) hypotenuse θ adjacent opposite In the following, each of the words opposite, adjacent, hypotenuse is the positive real number equal to the length of the correspondingly labeled side of the right triangle above. They depend only on the angle θ and not on the size of the triangle. The reason is that any two right triangles with angle θ are similar and the ratios of their corresponding sides are the same. Example. Find the six trigonometric ratios of the angle θ. θ How do you find trigonometric ratios for acute angle θ in a right triangle? sin θ = opposite cos θ = adjacent hypotenuse hypotenuse tan θ = opposite adjacent csc θ = hypotenuse opposite cot θ = adjacent opposite sec θ = hypotenuse adjacent The complete English names for these trig ratios are sine, cosine, tangent, cotangent, cosecant, and secant. All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6.: Right Triangles 8/5/6 Frame 6

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 6.. Right triangle trigonometric ratios (with Tamara Kucherenko) hypotenuse θ adjacent opposite In the following, each of the words opposite, adjacent, hypotenuse is the positive real number equal to the length of the correspondingly labeled side of the right triangle above. They depend only on the angle θ and not on the size of the triangle. The reason is that any two right triangles with angle θ are similar and the ratios of their corresponding sides are the same. Example. Find the six trigonometric ratios of the angle θ. Solution. sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = θ How do you find trigonometric ratios for acute angle θ in a right triangle? sin θ = opposite cos θ = adjacent hypotenuse hypotenuse tan θ = opposite adjacent csc θ = hypotenuse opposite cot θ = adjacent opposite sec θ = hypotenuse adjacent The complete English names for these trig ratios are sine, cosine, tangent, cotangent, cosecant, and secant. All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6.: Right Triangles 8/5/6 Frame 6

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 6.. Right triangle trigonometric ratios (with Tamara Kucherenko) hypotenuse θ adjacent opposite In the following, each of the words opposite, adjacent, hypotenuse is the positive real number equal to the length of the correspondingly labeled side of the right triangle above. How do you find trigonometric ratios for acute angle θ in a right triangle? sin θ = opposite cos θ = adjacent hypotenuse hypotenuse tan θ = opposite adjacent csc θ = hypotenuse opposite cot θ = adjacent opposite sec θ = hypotenuse adjacent They depend only on the angle θ and not on the size of the triangle. The reason is that any two right triangles with angle θ are similar and the ratios of their corresponding sides are the same. Example. Find the six trigonometric ratios of the angle θ. Solution. sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = Example. If sin θ = 4, sketch a right 7 triangle with acute angle θ. θ The complete English names for these trig ratios are sine, cosine, tangent, cotangent, cosecant, and secant. All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6.: Right Triangles 8/5/6 Frame 6

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 6.. Right triangle trigonometric ratios (with Tamara Kucherenko) hypotenuse θ adjacent opposite In the following, each of the words opposite, adjacent, hypotenuse is the positive real number equal to the length of the correspondingly labeled side of the right triangle above. How do you find trigonometric ratios for acute angle θ in a right triangle? sin θ = opposite cos θ = adjacent hypotenuse hypotenuse tan θ = opposite adjacent csc θ = hypotenuse opposite cot θ = adjacent opposite sec θ = hypotenuse adjacent They depend only on the angle θ and not on the size of the triangle. The reason is that any two right triangles with angle θ are similar and the ratios of their corresponding sides are the same. Example. Find the six trigonometric ratios of the angle θ. Solution. sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = Example. If sin θ = 4, sketch a right 7 triangle with acute angle θ. Solution. Since sin θ is defined as the ratio of the opposite side to the hypotenuse, we sketch a triangle with hypotenuse of length 7 and a side of length 4 opposite to θ. θ θ 7 4 The complete English names for these trig ratios are sine, cosine, tangent, cotangent, cosecant, and secant. All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6.: Right Triangles 8/5/6 Frame 6

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 6.. Right triangle trigonometric ratios (with Tamara Kucherenko) hypotenuse θ adjacent opposite In the following, each of the words opposite, adjacent, hypotenuse is the positive real number equal to the length of the correspondingly labeled side of the right triangle above. How do you find trigonometric ratios for acute angle θ in a right triangle? sin θ = opposite cos θ = adjacent hypotenuse hypotenuse tan θ = opposite adjacent csc θ = hypotenuse opposite cot θ = adjacent opposite sec θ = hypotenuse adjacent The complete English names for these trig ratios are sine, cosine, tangent, cotangent, cosecant, and secant. They depend only on the angle θ and not on the size of the triangle. The reason is that any two right triangles with angle θ are similar and the ratios of their corresponding sides are the same. Example. Find the six trigonometric ratios of the angle θ. Solution. sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = Example. If sin θ = 4, sketch a right 7 triangle with acute angle θ. Solution. Since sin θ is defined as the ratio of the opposite side to the hypotenuse, we sketch a triangle with hypotenuse of length 7 and a side of length 4 opposite to θ. To find the adjacent side we use the Pythagorean Theorem: (adjacent) = 7 4 =, so adjacent =. θ θ 7 4 All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6.: Right Triangles 8/5/6 Frame 6

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 6.. Special right triangles We can use the special triangles below to calculate the trigonometric ratios for angles with measures, 45, and 6. 45 45 6 θ θ rad sin θ cos θ tan θ csc θ sec θ cot θ 6 45 4 6 You do not need to, and should not, memorize this table. Rather, memorize the two red triangles and be able to quickly calculate all trig functions of, 45, or 6 degrees. In some situations, you are given some sides and angles of a triangle, and you want to find the other sides and angles. How to solve a triangle Find all side lengths and angles from the given information. Example : Solve the right triangle B A 5 Solution: The unknown angle is angle A. Since the acute angles in a triangle add to 9 = radians, angle A = = 6. The unknown sides are AC and BC. To find AC, we look for an equation that relates AC to the lengths and angles we already know. In this case, sin = AC AB = AC 5. Therefore AC = 5 sin = 5. Similarly, BC = 5 cos = 5. All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6.: Right Triangles 8/5/6 Frame 7 C

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 6.. Every real number is the radian measure of some angle Every real number, viewed as a number of radians, represents an angle. The initial side of the angle is a horizontal arrow pointing right from the origin. Rotate that arrow counterclockwise (or clockwise) with each full turn counting as (or ) radians. The final position of the arrow is the terminal side of the angle. At the end of this chapter, we will explain why our new definitions of cos θ, sin θ, and tan θ yield the same answers as the SohCahToa approach whenever θ is an acute angle. Y In the example at the right, the initial side is rotated through of a complete circle. The rotation is 4 suggested by the red dashed arc. The angle θ corresponds to of a complete trip around the circle, 4 and so θ = 4 = rad = 7. The main focus of high school trigonometry courses is the study of acute angles θ, those that satisfy < θ < 9. Every right triangle contains two acute angles. Trig functions (sine, cosine, tangent) of an acute angle are defined as ratios of the triangle s side lengths, abbreviated by the mnemonic SohCahToa. In this section we use a completely different method to define trig functions of general angles (measuring any real number of degrees or radians). This more general approach is crucial in science and engineering courses. θ terminal side initial side All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame 8

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions Radians / / /4 /6 Degrees 6 8 9 6 45 Y Y Y Y rad = 5 4 rad = 8 rad = 6 5 rad = 5 4 Y Y Y Y 5 rad = 5 4 rad = 8 rad = 6 - rad = 5 4 All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame 9

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions Trig functions of general angles We continue to view angles as arcs on a radius r circle, whose equation is x + y = r. If the angle θ is in standard position, its initial side is the ray from (, ) to (r, ). Suppose the terminal side is the ray from (, ) to point P (x, y) on the circle. We define trig functions of angle θ as follows. 5 Y 4 P (x, y) (4, ) The trig functions of angle θ cos(θ) = x r sec(θ) = cos θ = r x sin(θ) = y r csc(θ) = sin θ = r y tan(θ) = y x cot(θ) = tan θ = x y In the picture at the right, the arrow tip of the terminal side is at P (4, ). Thus x = 4, y =, r = x + y = 5 and so cos(θ) = x r = 4 5 sec(θ) = r x = 5 4 sin(θ) = y r = 5 csc(θ) = r y = 5 tan(θ) = y x = 4 cot(θ) = r y = 4 r = 5 θ -5-4 - - - 4 5 - - - -4-5 All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 5 Y The previous example used an angle θ with terminal side in Quadrant. The angle is approximately 7. and is called an acute angle because it satisfies < θ < / However, our definitions of trig functions apply equally well to any angle whatsoever. In the picture at the right, the arrow tip of the terminal side is at P ( 4, ). Thus x = 4, y =, r = x + y = 5 and so cos(θ) = x r = 4 5 sin(θ) = y r = 5 tan(θ) = y x = 4 and similarly for the other trig functions. ( 4, ) P (x, y) r = 5 4-5 -4 - - - 4 5 - - - -4 θ -5 All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions Trig functions of angles in Quadrants,,4 On this slide we are concerned not with the values of trig functions, but with their signs. Remember that r is always positive, while x and y can be positive or negative. cos(θ) = x/r has the same sign as x, which is positive in Q and Q4, negative in Q and Q sin(θ) = y/r has the same sign as y, which is positive in Q and Q, negative in Q and Q4. tan(θ) = y/x is positive in Q and Q, negative in Q and Q4. Summarizing these results by quadrant: The quadrant by quadrant results are: How to find signs of trig functions If the terminal side of the angle is in Q: x is +, y is +, so All cos, sin, tan are +. 5 Y 4 θ x -5-4 - - - 4 5 - - - -4-5 Quadrant P (x = 4, y = ) All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions Trig functions of angles in Quadrants,,4 On this slide we are concerned not with the values of trig functions, but with their signs. Remember that r is always positive, while x and y can be positive or negative. cos(θ) = x/r has the same sign as x, which is positive in Q and Q4, negative in Q and Q sin(θ) = y/r has the same sign as y, which is positive in Q and Q, negative in Q and Q4. tan(θ) = y/x is positive in Q and Q, negative in Q and Q4. Summarizing these results by quadrant: The quadrant by quadrant results are: How to find signs of trig functions If the terminal side of the angle is in Q: x is +, y is +, so All cos, sin, tan are +. Q: x is, y is +, so Sin is + but cos and tan are. Quadrant P (x = 4, y = ) 5 Y 4 x -5-4 - - - 4 5 - - - -4-5 θ All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions Trig functions of angles in Quadrants,,4 On this slide we are concerned not with the values of trig functions, but with their signs. Remember that r is always positive, while x and y can be positive or negative. cos(θ) = x/r has the same sign as x, which is positive in Q and Q4, negative in Q and Q sin(θ) = y/r has the same sign as y, which is positive in Q and Q, negative in Q and Q4. tan(θ) = y/x is positive in Q and Q, negative in Q and Q4. Summarizing these results by quadrant: The quadrant by quadrant results are: How to find signs of trig functions If the terminal side of the angle is in Q: x is +, y is +, so All cos, sin, tan are +. Q: x is, y is +, so Sin is + but cos and tan are. Q: x is, y is, so Tan is +, but cos and sin are. x -5-4 - - - 4 5 P (x = 4, y = ) Quadrant 5 Y 4 - - - -4-5 θ All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions Trig functions of angles in Quadrants,,4 On this slide we are concerned not with the values of trig functions, but with their signs. Remember that r is always positive, while x and y can be positive or negative. cos(θ) = x/r has the same sign as x, which is positive in Q and Q4, negative in Q and Q sin(θ) = y/r has the same sign as y, which is positive in Q and Q, negative in Q and Q4. tan(θ) = y/x is positive in Q and Q, negative in Q and Q4. Summarizing these results by quadrant: The quadrant by quadrant results are: How to find signs of trig functions If the terminal side of the angle is in Q: x is +, y is +, so All cos, sin, tan are +. Q: x is, y is +, so Sin is + but cos and tan are. Q: x is, y is, so Tan is +, but cos and sin are. Q4: x is +, y is, so Cos is +, but sin and tan are. Note that the first letters of the trig functions that are positive in Q,Q,Q,Q4 are A,S,T,C, sometimes remembered as All Students Take Calculus. 5 Y 4 x -5-4 - - - 4 5 - - - -4-5 θ 4 P 4 (x = 4, y = ) Quadrant 4 All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions We now show that trig functions of any angle can be easily computed from the corresponding trig functions of a closely related angle. Start with angle θ from before. It will reappear in the next pictures. cos(θ) = x r = 4 5 sin(θ) = y r = 5 5 Y 4 P(4,) tan(θ) = y x = 4 In this example, θ is a Quadrant angle. This means: θ is an angle whose terminal side lies in Quadrant. All trig functions of θ are positive. θ will reappear in the next slides as the reference angle of angles in Quadrants,, and 4. Definition The reference angle of a given angle is the acute angle between the x-axis and the given angle s terminal line. The reference angle θ is colored yellow on these slides. It is a positive angle, between and 9, and therefore is drawn as a counterclockwise arc. All of its trig functions are positive. θ x -5-4 - - - 4 5 - - - -4-5 r = 5 All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions How to find the reference angle of a Quadrant (Q) angle θ. If 9 < θ < 8 (/ < θ < ) then θ has reference angle θ = 8 θ = θ. How to find signs of trig functions of a Q angle θ cos(θ ) is negative since x is negative in Q. Sin(θ ) is positive since y is positive in Q. tan(θ ) is negative since y/x is negative in Q. These statements also follow from ASTC. How to find trig functions of a Q angle Suppose θ is in Quadrant and its reference angle is θ. cos(θ ) = cos(θ). sin(θ ) = + sin(θ). tan(θ ) = tan(θ). The endpoint of the terminal side of θ is P ( 4, ) Thus x = 4, y =, r = ( 4) + = 5. cos(θ ) = x r = 4 5 = 4 5 sin(θ ) = y r = 5 tan(θ ) = y x = 4 = 4 5 Y 4 P ( 4, ) θ r = 5 θ x -5-4 - - - 4 5 - - - -4-5 All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame 4

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions How to find the reference angle of a Quadrant (Q) angle θ. If 8 < θ < 7 ( < θ < /), then θ has reference angle θ = θ 8 = θ. How to find signs of trig functions of a Q angle θ cos(θ ) is negative since x is negative in Q. sin(θ ) is negative since y is negative in Q. Tan(θ ) is positive since y/x is positive in Q. These statements also follow from ASTC. How to find trig functions of a Q angle Suppose θ is in Quadrant and its reference angle is θ. cos(θ ) = cos(θ). sin(θ ) = sin(θ). tan(θ ) = + tan(θ). The endpoint of the terminal side of θ is P ( 4, ) Thus x = 4, y =, r = ( 4) + ( ) = 5. cos(θ ) = x r = 4 5 = 4 5 sin(θ ) = y r = 5 = 5 tan(θ ) = y x = 4 = 4 tan(θ ) = + tan(θ). -5-4 - - - 4 5 x θ r = 5 - - - P ( 4, ) -4-5 5 Y 4 θ All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame 5

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions How to find the reference angle of a Quadrant 4 (Q4) angle. If 7 < θ 4 < 6 (/ < θ 4 < ), then θ 4 has reference angle θ = 6 θ 4 = θ 4. How to find signs of trig functions of a Q4 angle θ 4 cos(θ 4 ) is positive since x is positive in Q4. sin(θ 4 ) is negative since y is negative in Q4. tan(θ 4 ) is negative since y/x is negative in Q4. These statements also follow from ASTC. How to find trig functions of a Q4 angle Suppose θ 4 is in Quadrant 4 and its reference angle is θ. cos(θ 4 ) = + cos(θ). sin(θ 4 ) = sin(θ). tan(θ 4 ) = tan(θ). 5 Y 4 θ 4-5 -4 - - - 4 5 x θ - r = 5 The endpoint of the terminal side of θ 4 is P 4 (4, ) Thus x = 4, y =, r = 4 + ( ) = 5. cos(θ 4 ) = x r = 4 5 = 4 5 sin(θ 4 ) = y r = 5 = 5 tan(θ 4 ) = y x = 4 = 4 - - -4-5 P 4 (4, ) All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame 6

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 6..4 Finding the reference angle of a given angle. We abbreviate the reference angle of θ by Ref(θ ). It is the angle θ shown in yellow on the previous slides. Ref(θ ) is the smaller angle between the terminal line of θ and the x-axis. The reference angle is always positive: <Ref(θ ) < 9 Similar statements hold for the angles θ and θ 4. How to find the degree measure of the reference angle Q: If < θ < 9 then Ref(θ ) = θ. Q: If 9 < θ < 8 then Ref(θ ) = 8 θ. Q: If 8 < θ < 7 then Ref(θ ) = θ 8. Q4: If 7 < θ 4 < 6 then Ref(θ 4 ) = 6 θ 4. Angles, ±9, ±8... that are multiples of 9 have undefined reference angles. If θ < or θ > 6, add or subtract a multiple of 6 to/from θ to get an angle θ with θ < 6. Then Ref(θ) = Ref(θ ), computed by the above rules. Example : Find the cosines of the following angles: 5, 5, Solution: 5 : Terminal line in Q: Ref(5 ) = 8 5 =. Since cosine is negative in Q, cos(5 ) = cos( ) = 5 : Terminal line in Q: Ref(5 ) = 5 8 = 45. Since cosine is negative in Q, cos(5 ) = cos(45 ) = : Terminal line in Q4: Ref( ) = 6 = 6. Since cosine is positive in Q4, cos( ) = + cos(6 ) = How to find the radian measure of the reference angle Q: If < θ < then Ref(θ ) = θ. Q: If < θ < then Ref(θ ) = θ. Q: If < θ < then Ref(θ ) = θ. Q4: If < θ 4 < then Ref(θ 4 ) = θ 4. Angles, ±, ±, ±... that are multiples of have undefined reference angles. If θ < or θ >, add or subtract a multiple of to/from θ to get an angle θ with θ <. Then Ref(θ) = Ref(θ ), computed by the above rules. All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame 7

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions The following summary need not be memorized if you understand the previous slides. Given any angle θ, let Ref(θ) be the reference angle of θ. Trig functions of angles in terms of the reference angle cos θ = cos Ref(θ) for angles θ ending in Q and Q4. cos θ = cos Ref(θ) for angles θ ending in Q and Q. sin θ = sin Ref(θ) for angles θ ending in Q and Q. sin θ = sin Ref(θ) for angles θ ending in Q and Q4. tan θ = tan Ref(θ) for angles θ ending in Q and Q. tan θ = tan Ref(θ) for angles θ ending in Q and Q4. Example : Find the reference angle of 9. Solution: 9 6 6 =. Since 8 < < 7, angle is in Q. Its reference angle is 8 =. Example : Find the reference angle of 9. Solution: Method : Convert to degrees: 9( ) = 9(6 ) = 4. Then 4 6 6 6 = 6 is in Q, and so its reference angle is 6. Method : Keep radians. 9 9 = is in Q and so its reference angle is 6 = 9 8. = All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame 8

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 6..5 Trig functions generalize SohCahToa The diagram below shows an acute triangle in Quadrant. The terminal side of angle θ goes from the origin to P (x, y), while the the distance from P to the origin is r = x + y. The red arc shows the angle θ in standard position with terminal line OP. Here s the crucial point: because x, y, and r are positive numbers, they are the side lengths of the triangle. x is the length of the side adjacent to angle θ. y is the length of the side opposite angle θ. r is the length of the hypotenuse. Our definitions of trig functions are as follows: cos(θ) = x r = sec(θ) = r x = hypotenuse adjacent adjacent sin(θ) = y hypotenuse r = opposite hypotenuse sec(θ) = r y = hypotenuse opposite tan(θ) = y x = opposite adjacent cot(θ) = x y = adjacent opposite 5 Y The above formulas define trig functions of all angles θ (for < θ < ) agree with the SohCahToa definitions of trig functions of acute angles θ ( < θ < ). 4 P (x, y) θ r y θ x x 4 5 All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame 9

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 6..6 Pythagorean trigonometric identities We just saw that x + y = r. Now it s easy to obtain the three basic trig identities. Dividing x + y = r by r gives ( ) x ( r + y ) ( r = r ) : r Therefore cos θ + sin θ = Solve for each trig function to obtain cos(θ) = ± sin θ and sin(θ) = ± cos θ by x gives ( ) x ( x + y ) ( x = r ) : x Therefore + tan θ = sec θ Solve for each trig function to obtain tan θ = ± sec θ and sec θ = ± + tan θ ( ) ( ) ( ) by y gives x y + y y = r : y Therefore cot θ + = csc θ Solve for each trig function to obtain cot(θ) = ± csc θ and csc(θ) = ± + cot θ In all cases, the choice of plus or minus depends on the quadrant of angle θ: Example 4: If θ is a Quadrant angle, express tan(θ) in terms of sin(θ). Solution: Since tan θ = sin θ, and the goal is to cos θ express tan θ in terms of sin θ, solve for cos θ in cos θ + sin θ = to obtain cos θ = ± (sin θ). Since θ is in quadrant : sin θ is negative and tan θ is positive. Then tan θ = sin θ cos θ = sin θ ± tells us: (sin θ) negative Positive = ± (sin θ) ; Therefore ± (sin θ) is negative. sin θ tan θ = sin θ = sin θ sin θ Check your answer: In Quadrant, sin θ is negative while tan θ is positive. Therefore the answers in the box are both positive. If they would have different signs, you would need to go back and check for an error. All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 6..7 Evaluating trigonometric functions Example 5: If sec(θ) = 5 and θ is in Quadrant, find the other 5 trig functions of θ. Solution: If the terminal line of θ goes from the origin to P (x, y), located distance r from the origin, then sec(θ) = r x = 5. Choose any r and x that satisfy this equation, but remember that r is positive. The easiest solution is r = 5 and x =. But x + y = r. Thus ( ) + y = 5 9 + y = 5 and so y = ±4. But (x, y) is in quadrant, where y is positive, and so y = 4. Therefore x =, y = 4, r = 5. Note that sec(θ) = r x = 5 checks with the given information. The requested five trig functions are cos(θ) = x r = 5 = 5 sin(θ) = y r = 4 5 5 4 P (, 4) tan(θ) = y x = 4 = 4 cot(θ) = x y = 4 = 4 csc(θ) = r y = 5 4 r = 5 Y θ -5-4 - - - 4 5 x All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 6..8 How your calculator computes cosines Example 6: If θ is the radian measure of an angle, it is shown in Calculus III that if / θ /, then an approximate formula for cosine is cos(θ) θ! + θ4 4! θ6 6! You know that cos(/) =. Test the accuracy of the above formula with θ = /.47. Solution: Compute by hand with decimal place accuracy, or (if you must) use a calculator to check that.47 +.474 4.476 7.5 All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions The functions y = cos(θ) and y = θ! + θ4 4! θ6 are graphed below. 6! Note that the graphs look identical for θ - y = cos(θ) y = θ! + θ4 4! θ6 6! All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions How close to identical, really? The difference of the functions y = cos(θ) and y = θ graphed below. Look at the scale to the left of the graph! When θ = /, the difference equals.54467. The above simple formula gives the correct answer with better than 99.99 per cent accuracy!.! + θ4 4! θ6 is 6! y = θ! + θ4 4! θ6 6! cos(θ) -. All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame 4

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 6. Quiz Review Example : Find the cosines of the following angles: 5, 5, Example : Find the reference angle of 9. Example : Find the reference angle of 9. Example 4: If θ is a Quadrant angle, express tan(θ) in terms of sin(θ). Example 5: If sec(θ) = 5 and θ is in Quadrant, find the other 5 trig functions of θ. All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 6. Trigonometric functions of angles 7/5/6 Frame 5

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 5.. Graphing sin θ as a function of angle θ If the endline of angle θ joins the origin to point (x, y) on the circle x + y =, then cos θ = x and sin θ = y So to start drawing the graph of Y = sin θ for θ,travel around the circle x + y = as follows. Start at point (, ) (corresponding to θ = ), move counterclockwise around the circle until you return to your starting point. The following landmarks in your trip occur each quarter-circle turn: As θ goes from: On the circle x + y = point On the graph of y = sin θ (x, y) = (cos θ, sin θ) goes from (θ, sin θ) goes from to / (, ) to (, ) (, ) to (/, ) Y Y (/, ) Local and absolute maximum - All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 5.: Trigonometric graphs 8/5/6 Frame 6

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 5.. Graphing sin θ as a function of angle θ If the endline of angle θ joins the origin to point (x, y) on the circle x + y =, then cos θ = x and sin θ = y So to start drawing the graph of Y = sin θ for θ,travel around the circle x + y = as follows. Start at point (, ) (corresponding to θ = ), move counterclockwise around the circle until you return to your starting point. The following landmarks in your trip occur each quarter-circle turn: As θ goes from: On the circle x + y = point On the graph of y = sin θ (x, y) = (cos θ, sin θ) goes from (θ, sin θ) goes from to / (, ) to (, ) (, ) to (/, ) / to (, ) to (, ) (/, ) to (, ) Y Y (/, ) Local and absolute maximum - All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 5.: Trigonometric graphs 8/5/6 Frame 6

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 5.. Graphing sin θ as a function of angle θ If the endline of angle θ joins the origin to point (x, y) on the circle x + y =, then cos θ = x and sin θ = y So to start drawing the graph of Y = sin θ for θ,travel around the circle x + y = as follows. Start at point (, ) (corresponding to θ = ), move counterclockwise around the circle until you return to your starting point. The following landmarks in your trip occur each quarter-circle turn: As θ goes from: On the circle x + y = point On the graph of y = sin θ (x, y) = (cos θ, sin θ) goes from (θ, sin θ) goes from to / (, ) to (, ) (, ) to (/, ) / to (, ) to (, ) (/, ) to (, ) to / (, ) to (, ) (, ) to (/, ) Y Y (/, ) Local and absolute maximum - (/,-) Local and absolute minimum All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 5.: Trigonometric graphs 8/5/6 Frame 6

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 5.. Graphing sin θ as a function of angle θ If the endline of angle θ joins the origin to point (x, y) on the circle x + y =, then cos θ = x and sin θ = y So to start drawing the graph of Y = sin θ for θ,travel around the circle x + y = as follows. Start at point (, ) (corresponding to θ = ), move counterclockwise around the circle until you return to your starting point. The following landmarks in your trip occur each quarter-circle turn: As θ goes from: On the circle x + y = point On the graph of y = sin θ (x, y) = (cos θ, sin θ) goes from (θ, sin θ) goes from to / (, ) to (, ) (, ) to (/, ) / to (, ) to (, ) (/, ) to (, ) to / (, ) to (, ) (, ) to (/, ) / to (, ) to (, ) (/, ) to (, ) Y Y (/, ) Local and absolute maximum y = sin(x) - (/,-) Local and absolute minimum All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 5.: Trigonometric graphs 8/5/6 Frame 6

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions Graphing cos θ as a function of angle θ If the endline of angle θ joins the origin to point (x, y) on the circle x + y =, then cos θ = x and sin θ = y So to start drawing the graph of Y = cos θ for θ,travel around the circle x + y = as follows. Start at point (, ) (corresponding to θ = ), move counterclockwise around the circle until you return to your starting point. The following landmarks in your trip occur each quarter-circle turn: As θ goes from: On the circle x + y = point On the graph of y = cos θ (x, y) = (cos θ, sin θ) goes from (θ, cos θ) goes from to / (, ) to (, ) (, ) to (/, ) Y Y (, ) Absolute maximum - All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 5.: Trigonometric graphs 8/5/6 Frame 7

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions Graphing cos θ as a function of angle θ If the endline of angle θ joins the origin to point (x, y) on the circle x + y =, then cos θ = x and sin θ = y So to start drawing the graph of Y = cos θ for θ,travel around the circle x + y = as follows. Start at point (, ) (corresponding to θ = ), move counterclockwise around the circle until you return to your starting point. The following landmarks in your trip occur each quarter-circle turn: As θ goes from: On the circle x + y = point On the graph of y = cos θ (x, y) = (cos θ, sin θ) goes from (θ, cos θ) goes from to / (, ) to (, ) (, ) to (/, ) / to (, ) to (, ) (/, ) to (, ) Y Y (, ) Absolute maximum - All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 5.: Trigonometric graphs 8/5/6 Frame 7

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions Graphing cos θ as a function of angle θ If the endline of angle θ joins the origin to point (x, y) on the circle x + y =, then cos θ = x and sin θ = y So to start drawing the graph of Y = cos θ for θ,travel around the circle x + y = as follows. Start at point (, ) (corresponding to θ = ), move counterclockwise around the circle until you return to your starting point. The following landmarks in your trip occur each quarter-circle turn: As θ goes from: On the circle x + y = point On the graph of y = cos θ (x, y) = (cos θ, sin θ) goes from (θ, cos θ) goes from to / (, ) to (, ) (, ) to (/, ) / to (, ) to (, ) (/, ) to (, ) to / (, ) to (, ) (, ) to (/, ) Y Y (, ) Absolute maximum -,-) Absolute minimum All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 5.: Trigonometric graphs 8/5/6 Frame 7

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions Graphing cos θ as a function of angle θ If the endline of angle θ joins the origin to point (x, y) on the circle x + y =, then cos θ = x and sin θ = y So to start drawing the graph of Y = cos θ for θ,travel around the circle x + y = as follows. Start at point (, ) (corresponding to θ = ), move counterclockwise around the circle until you return to your starting point. The following landmarks in your trip occur each quarter-circle turn: As θ goes from: On the circle x + y = point On the graph of y = cos θ (x, y) = (cos θ, sin θ) goes from (θ, cos θ) goes from to / (, ) to (, ) (, ) to (/, ) / to (, ) to (, ) (/, ) to (, ) to / (, ) to (, ) (, ) to (/, ) / to (, ) to (, ) (/, ) to (, ) Y Y (, ) Absolute maximum (, ) Absolute maximum y = cos(x) -,-) Absolute minimum All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 5.: Trigonometric graphs 8/5/6 Frame 7

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 5.. Graphing y = sin x Y - Example : Sketch the graph of y = sin(x) for x. Plot the five blue points: sin(x) = at x =,, ; sin( ) = ; sin( ) = Start at (,) All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 5.: Trigonometric graphs 8/5/6 Frame 8

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 5.. Graphing y = sin x Y (/, ) Local and absolute maximum - Example : Sketch the graph of y = sin(x) for x. Plot the five blue points: sin(x) = at x =,, ; sin( ) = ; sin( ) = Start at (,) Go up to point (/,), a local maximum because it is at the top of a hill. All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 5.: Trigonometric graphs 8/5/6 Frame 8

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 5.. Graphing y = sin x Y (/, ) Local and absolute maximum - Example : Sketch the graph of y = sin(x) for x. Plot the five blue points: sin(x) = at x =,, ; sin( ) = ; sin( ) = Start at (,) Go up to point (/,), a local maximum because it is at the top of a hill. Down to (, ) All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 5.: Trigonometric graphs 8/5/6 Frame 8

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 5.. Graphing y = sin x Y (/, ) Local and absolute maximum - (/,-) Local and absolute minimum Example : Sketch the graph of y = sin(x) for x. Plot the five blue points: sin(x) = at x =,, ; sin( ) = ; sin( ) = Start at (,) Go up to point (/,), a local maximum because it is at the top of a hill. Down to (, ) Down to point (/,-), a local minimum because it is at the bottom of a valley. All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 5.: Trigonometric graphs 8/5/6 Frame 8

Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions 5.. Graphing y = sin x Y - (/, ) Local and absolute maximum y = sin(x) (/,-) Local and absolute minimum Example : Sketch the graph of y = sin(x) for x. Plot the five blue points: sin(x) = at x =,, ; sin( ) = ; sin( ) = Start at (,) Go up to point (/,), a local maximum because it is at the top of a hill. Down to (, ) Down to point (/,-), a local minimum because it is at the bottom of a valley. Back up to (, ). All rights reserved. Copyright 6 by Stanley Ocken CCNY Math Review Chapter 5.: Trigonometric graphs 8/5/6 Frame 8