Chapter 4: Trigonometry Section 4-1: Radian and Degree Measure INTRODUCTION An angle is determined by rotating a ray about its endpoint. The starting position of the ray is the of the angle, and the position after rotation is the, as shown. The endpoint of the ray is the of the angle. This perception of an angle fits a coordinate system in which the is the vertex and the initial side coincides with the positive x-axis. Such an angle is in standard position, as shown. Positive angles are generated by and negative angles by, as shown. In this figure, note that angles α and β have the same initial and terminal sides. Such angles are coterminal. 1
In your own words: Coterminal: Standard position: RADIANS The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. One way to measure angles is in ; the other way is in radians. This type of measure is especially useful in calculus. One revolution around a circle, degrees, equals 2π radians. Thus, 1/2 revolution = radians 1/4 revolution = radians 1/6 revolution = radians Other common angles are shown below: 2
Label the following coordinate system with Quadrants I, II, III, and IV. Then, determine which angle intervals, in radians, fall in each quadrant. Also label the axes in radians. FINDING COTERMINAL ANGLES To find coterminal angles, add or subtract 2π from the angle. For example: 0 and are coterminal angles. 6 and are coterminal angles. Example 1: Sketching and Finding Coterminal Angles a. Find two coterminal angles for 13π/6 and sketch them. b. Find a positive and a negative coterminal angle for 3π/4 and sketch them. c. Find two positive coterminal angles for -2π/3 and sketch them. 3
COMPLEMENTARY AND SUPPLEMENTARY ANGLES Remember from Geometry that complementary angles have a sum of, which is radians and supplementary angles have a sum of, which is radians. Example #2: Complementary and Supplementary Angles If possible, find the complement and supplement of a) 2 /5 and b) 4π/5. DEGREE MEASURE Another way to measure angles is in terms of degrees. 1 is equivalent to 1/360 of a complete revolution about the vertex. Thus, a full revolution corresponds to 360, a half revolution to 180, and a quarter revolution to 90. Because 2π radians corresponds to one complete revolution, degrees and radians are related by the equations 360 = rad and = π rad 4
CONVERSIONS BETWEEN DEGREES AND RADIANS 1) To convert degrees to radians, multiply degrees by 180 2) To convert radians to degrees, multiply radians by 180 Example #3: Converting from Degrees to Radians a) 135 b) 540 c) -270 Example #4: Converting from Radians to Degrees a) - 2 rad = b) 9 2 rad = c) 2 rad = 5
APPLICATIONS The radian measure formula can be used to measure arc length along a circle. For a circle of radius r, a central angle θ intercepts an arc of length s given by s = r θ where θ is measured in radians. Example #5: Finding Arc Length A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240. Consider a circle of radius r. If s is the length of the arc traveled in time t, the speed of the particle is Speed = = Moreover, if θ is the angle (in radian measure) corresponding to the arc length s, the angular speed of the particle is Angular speed = 6
Example #6: Finding the Speed of an Object The second hand of a clock is 10.2 centimeters long. Find the speed of the tip of this second hand. Example #7: Finding Angular Speed A lawn roller makes 1.2 revolutions per second. Find the angular speed of the roller in radians per second. 7
Section 4-2: Trigonometric Functions: The Unit Circle The Unit Circle: Graph the circle x 2 + y 2 = 1. Now imagine the real number line is wrapped around this circle, with numbers corresponding to a counterclockwise wrapping and numbers corresponding to a clockwise wrapping. The real number 0 corresponds to the point (1, 0). Moreover, because the unit circle has a circumference of 2π, the real number 2π corresponds to the point (, ). Why is the x value 1? The Trigonometric Functions: Recall sine, cosine and tangent from right angle trigonometry. In geometry, Sine = Cosine = Tangent = Three more functions cosecant, secant and cotangent, respectively combine with these 3 to make the 6 trigonometric functions. Definitions of Trigonometric Functions Let t be a real number and let (x, y) be the point on the unit circle corresponding to t. sin t = y csc t =, y 0 cos t = x sec t =, x 0 tan t =, x 0 cot t =, y 0 Note: The functions in the second column are the of the corresponding functions in the first column. 8
Recalling right angle trigonometry, why does sin t = y? cos t = x? tan t =? In the definitions of trigonometric functions, note that tangent and secant are not defined when x = 0. Thus, because t = corresponds to (x, y) = (, ), it follows that tan ( ) and sec ( ) are undefined. Similarly, and are not defined when y = 0. For instance, because t = 0 corresponds to (x, y) = (, ), and are undefined. Let s create the unit circle. Make sure you have a piece of graph paper, a piece of lined paper, 1) Graph the circle from the beginning of this section (x 2 + y 2 = 1) nearly as large as the paper will allow you. 2) Sketch in all angles for all multiples of 30 and 45 from 0 to 360. Label near the origin, inside the circle. 3) Compute all conversions (from degrees to radians on lined paper) for all multiples of 30 and 45 from 0 to 360. 4) Label each angle in radians on graph paper. 5) On lined paper, draw the first triangle in the unit circle (θ = 30 or radians). Label the hypotenuse. 6) Compute the opposite side and adjacent side (recall 30-60 -90 and 45-45 -90 triangles leave in radical form). 7) Label the point on the unit circle corresponding to the first angle as an ordered pair with the side as the x coordinate and the side as the y coordinate. 8) Repeat for every angle sketched. 9
Example #1: Evaluating Trigonometric Functions Evaluate the six trigonometric functions at each real number. a) t = π 6 b) t = 5π 4 c) t = 0 d) t =π 10
Example #2: Evaluating Trigonometric Functions Evaluate the six trigonometric functions at t = π 3. Domain and Period of Sine and Cosine The domain of the sine and cosine functions is the set of all real numbers. To determine the range of these two functions, consider the unit circle. Because r = 1, it follows that sin t = y and cos t = x. Moreover, because (x, y) is on the unit circle, you know that 1 y 1 and 1 x 1, and it follows that the values of sine and cosine range between -1 and 1. That is, 1 y 1 and 1 x 1 1 sint 1 1 cost 1 Suppose you add 2π to each value of t in the interval 0,2π completing a second revolution around the unit circle. This leads to the general result sin(t+2πn)=sint and cos(t+2πn)= cost., thus Functions that behave in such a repetitive (or cyclic) manner are called periodic. 11
Definition of Periodic Function: A function f is periodic if there exists a positive real number c such that f (t+c)= f (t) for all t in the domain of f. The smallest number c for which f is periodic is called the period of f. Example #3: Using the Period to Evaluate the Sine and Cosine a) Evaluate sin 13π 6 b) Evaluate cos 7π 2 12
Recall from Section 1.4 that a function f is even if f ( t)= f (t) and is odd if f ( t)= f (t). Even and Odd Trigonometric Functions The cosine and secant functions are even. cos( t)= cos(t) and sec( t)=sec(t) The sine, cosecant, tangent, and cotangent functions are odd. sin( t)= sin(t) and csc( t)= csc(t) tan( t)= tan(t) and cot( t)= cot(t) Example #4: Using a Calculator Use a calculator to evaluate each expression. a) sin76.4 b) cot1.5 13
Section 4-3: Right Angle Trigonometry The Six Trigonometric Functions: Example 1: Evaluating Trigonometric Functions Find the values of the six trigonometric functions of θ as shown. Example #2: Evaluating Trigonometric Functions of 45 Find the values of sin 45, cos 45, and tan 45. Example #3: Evaluating Trigonometric Functions of 60 and 30 Use the equilateral triangle shown to find the values of sin 60, cos 60, sin 30, and cos 30 14
Sines, Cosines, and Tangents of Special Angles sin 30 = sin = sin 45 = sin = sin 60 = sin = cos 30 = cos = cos 45 = cos = cos 60 = cos = tan 30 = tan = tan 45 = tan = 1 tan 60 = tan = 3 Note that sin 30 = = cos 60. This occurs because 30 and 60 are angles, and, in general, confunctions of complementary angles are equal. That is, if θ is an acute angle, the following relationships are true: sin (90 - θ) = cos θ tan (90 - θ) = cot θ sec (90 - θ) = csc θ cos (90 - θ) = sin θ cot (90 - θ) = tan θ csc (90 - θ) = sec θ Trigonometric Identities: Fundamental Trigonometric Identities Quotient Identities tan θ = cot θ = Pythagorean Identities sin 2 θ + cos 2 θ = 1 1 + tan 2 θ = sec 2 θ 1 + cot 2 θ = csc 2 θ 15
Example #4: Applying Trigonometric Identities Let θ be an acute angle such that sin θ = 0.6. Find the values of (a) cos θ and (b) tan θ using trigonometric identities. Example #5: Applying Trigonometric Identities Let θ be an acute angle such that tan θ = 3. Find the values of (a) cot θ and (b) sec θ using trigonometric identities. 16
Applications Involving Right Triangles: Example #6: Solving a Right Triangle A surveyor is standing 50 feet from the base of a large tree, as shown. The surveyor measures the angle of elevation to the top of the tree as 71.5. How tall is the tree? Example #7: Solving a Right Triangle A person is 200 yards from a river. Rather than walking directly to the river, the person walks 400 yards along a straight path to the river s edge. Find the acute angle θ between this path and the river s edge, as illustrated. 17
Example #8: Solving a Right Triangle A 12-meter flagpole casts a 9-meter shadow, as shown. Find θ, the angle of elevation of the sun. 18