Unit 4 Trigonometr Stud Notes 1 Right Triangle Trigonometr (Section 8.1) Objective: Evaluate trigonometric functions of acute angles. Use a calculator to evaluate trigonometric functions. Use trigonometric functions to model and solve real life problems. Identif situations involving an angle of elevation and an angle of depression. B the End of Class I Should Mini Lesson on Rationalizing the Denominator: There is an agreement in mathematics that we don t leave a radical in the denominator of a fraction. To remove the radical from the denominator we multipl the numerator and the denominator b the radical. DON T FORGET TO SIMPLIFY! Eample 1: Simplif the following epressions. A. B. C. Trigonometr means measurement of angles. The easiest angles to deal with in trigonometr are the angles in right triangles. A right triangle is a triangle which has one right angle (90 ) and two acute angles (less than 90 each). In the right triangle below, one acute angle is named (theta), and with respect to that angle, ou need to be able to identif the opposite side, the adjacent side, and the hpotenuse of the triangle. Basic Right Triangle Trigonometr: The basic right triangle trigonometric ratios are given b Primar Functions Reciprocal Functions sine sin ( ) cosecant csc = cosine cos ( ) secant sec tangent tan ( ) cotangent cot = = When using right triangle trigonometr, angles are usuall measured in degrees. Eample 2: Use the triangle below to find the eact values of the si trigonometric functions of. 13 5 sin cos csc sec 12 tan cot
There are two tpes of special right triangles for which ou should memorize the pattern of side lengths, this will help build trig ratios. These are 45 45 90 and 30 60 90 triangles. Eample 3: Fill in the sides of the triangles below. Then find the trig ratios for each acute angle. sin 45 sin 60 sin 30 45 cos 45 30 cos 60 cos 30 45 tan 45 60 tan 60 tan 30 It is possible to find eact values of trig ratios when using special triangles or right triangles with two given sides. However, approimate values of trig ratios ma be needed to solve real world problems. Your calculator will find approimate trig ratios in decimal form. Make sure that ou have our calculator set in the correct mode. (Degree or Radian) (We will learn more about radians later). Eample 4: Find the following trig ratios using our calculator. (Make sure that ou are in degree mode). A. sin 57 B. cos 24 C. tan 71 D. csc 40 E. sec 12 F. cot 63 Eample 5: Solve for the missing part labeled for each triangle. A. B. 6 22 40 20 Think About It: What is the difference in solving for a side versus solving for an angle?
Eample 6: If tan, find the following values. A. cot B. sin C. sec Hint: If ou are ever not sure where to start DRAW A TRIANGLE!! Eample 7: Given the point 4, 6 is on the terminal side of an angle in standard position. Determine the eact values of the cosine, cosecant, and cotangent trigonometric functions of the angle. An angle of elevation is an angle measured from a horizontal upward toward an object. Observer Object Angle of Elevation An angle of depression is an angle measured from a horizontal downward toward an object. Observer Angle of Depression Object Eample 8: A biologist wants to know the width (w) of a river in order to properl set instruments for studing the pollutants in the water. From Point A, the biologist walks downstream 70 feet and sights to point C. From this sighting, it is determined that =54. How wide is the river? C 54 = A 70 ft. Biologist Eample 9: A sonar operator on a ship detects a submarine at a distance of 500 meters from the ship and an angle of depression of 40. How deep is the submarine?
Stud Notes 2 Trigonometric Functions of an Angle & the Unit Circle (Section 8.2 & 8.3) Objective: Use the unit circle to evaluate trigonometric functions. Use a calculator to evaluate trigonometric function for an angle. Find co terminal angles and their corresponding trig ratios. B the End of Class I Should The previous notes dealt with trigonometric functions for acute angles. This section etends the trigonometric functions to an angle b using reference angles and reference triangles. A discussion of angles (and their measures) in the coordinate plane is an important prerequisite to finding trig ratios for all possible angles. An angle is said to be in standard position in the coordinate plane if it is formed b a rotation from the positive ais. Initial side: Terminal side: Positive angles: Negative angles: The reference angle for an angle in standard position is the acute angle formed b its terminal side and the horizontal ais ( ais) Eample 1: Sketch angles in standard position having the following measures, and find their reference angles. A. 120 B. 315 C. 150 Reference Angle: Reference Angle: Reference Angle:
The Unit Circle is the circle centered at the origin with radius 1 unit (hence, the unit circle). The equation of this circle is 1. A diagram of the unit circle is shown. We are going to deal primaril with special angles around the unit circle, namel the multiples of 30, 45, 60, and 90. All angles throughout this unit will be drawn in standard position. Label Coordinates o Quadrant Angles Label Angles o Multiples of 90, Multiples of 45, Multiples of 60, Multiples of 30 Unit Circle
Labeling Coordinates for 30, 45, and 60. In standard position sketch a 45 reference angle, and construct a triangle based on the reference angle. Label the coordinates for the point, on the circle that was created b the reference angle of 45. Then repeat the process for 30 and 60. Label the 1 st quadrant of the unit circle with the coordinate points, for 45, 30, and 60. Now using our knowledge of the coordinate plane, fill in the coordinates for quadrants, II, III, IV. Let s go back to are si trigonometric functions and see how the relate to the unit circle.
Eample 2: Find the eact values of the following trigonometric functions: A. Find sin 60 B. cos 90 C. sin 45 D. tan 30 E. csc135 F. cos270 G. cos45 H. sec 150 I. sin 300 J. tan 60 K. csc 150 ) M. cot 120 N. cot 225 O. sin330 P. csc 180 Q. cos 240 ) R. tan 180 S. csc 270 T. sin 90 When angles are not available on the unit circle, use a calculator to evaluate the trigonometric function. Eample 3: Use a calculator to find the following: A. sin237 B. cos612 C. csc112 D. cot305
If angles in standard position share the same terminal side, the are called co-terminal angles. The angles in the diagram below have measures of 30, 390, and 330. The are co terminal. 30 330 390 Co terminal angles have measures which differ b multiples of 360. Thus, angles which are coterminal to a given angle can be found b adding or subtracting 360 as man times as desired. Eample 4: Sketch each angle in standard position, and find one negative and one positive co terminal angle for each. A. 120 B. 405 Since co terminal angles share the same terminal side, the form the same reference angles (and reference triangles). Thus, the have the same trig ratios. Eample 5: Find the eact values of the following trigonometric functions: A. sin 420 B. cos 585 C. tan 390 D. csc 660 E. sec 450 F. cot 480 Eample 6: Given that sin 31.5150, sin 51.7771, and sin 71.9455 (to 4 decimal place accurac), find the following without using a calculator. A. sin431 B. sin329 C. sin771
Stud Notes 3 Radian Measure for Angles (Section 8.4) Objective: Use radian measure for angles. Convert between radian and degree measure. B the End of Class I Should We have usuall learned to measure an angle in degrees. However, there are other units of angle measurement. One wa that we are going to look at is radians. In man scientific and engineering calculations, radians are used in preference to degrees. The arc shown has a length chosen equal to the radius; the angle is then 1 radian. B etension an angle of 2 radians will be subtended b an arc of length 2r. Notice that the length of the arc is alwas given b: In the general case, the arc length s, is found b So for a full circle the arc length is the same as its circumference, 2. Thus, 2 2 In other words, when we are working in radians, the angle in a full circle is 2 radians, thus This enables us to have a set of equivalences between degrees and radians. 360 2 180 90 2 45 4 60 3 30 6 **Finishing Labeling the Unit Circle**
Eample 1: Find the eact values of the following trigonometric functions: A. sin B. cos C. tan D. sec E. cot F. csc G. cot H. sec I. cos Converting Between Radians and Degrees: Radians to degrees, multipl b Degrees to radians, multipl b Eample 2: Convert from degrees to radians or vice versa. Do not use a calculator. Simplif all solutions. A. 240 B. C. 210 D. 4 (radians) When finding co-terminal angles for angles epressed in radian measure, add or subtract 2π as man times needed. (Remember not to add 360, as 360 is not epressed in radian measure). Eample 3: Sketch each angle in standard position, and find one positive and one negative co terminal angle for each. (Epress our answers in radian measure) A. B. 7 3 Eample 4: Use our calculator to find the following to 3 decimal place accurac. Check our mode. Note: When degrees are not specificall indicated, angle measures are considered to be in radians. A. cos 3.725 B. tan 4 C. csc 2.621 7
Stud Notes 4 Solving Trig Equations using Inverse Trigonometr (Section 8.5) Objective: Find angles from given trigonometric ratios. Solve trigonometric equations with a calculator. Solve real world problems b finding angles from given trigonometric ratios. B the End of Class I Should So far, ou have worked with trigonometric functions in a forward direction. You have been given angles, and ou have been asked to find trigonometric ratios (in either fractional or decimal form). Suppose ou were given a trig ratio and asked to find angles which produced that ratio. This is working in a backward direction. Man problems in the real world involve using trigonometr in a backward direction (ratios to angles), this is known as inverse operations. Eample 1: Solve for. 7 2 Inverse Operation on Your Calculator Use when finding an ANGLE Eample 2: A swimming pool is 20 meters long and 12 meters wide. The bottom of the pool is slanted such that the water depth is 1.3 meters at the shallow end and 4 meters at the deep end. Find the angle of depression of the bottom of the pool. 20 m 1.3 m 2.7 m The unit circle can also be used to work backwards. The question might be worded something like this: Find all angles, in the interval 0, 360, that satisf the given trigonometric equation. Think About It: Wh is a restriction given on the angle, 0, 360? When the restriction is in DEGREES our answer is in DEGREES! When the restriction is in RADIANS our answer is in RADIANS! Eample 3: Find all angles, in the interval 0, 360, that satisf cos. Where is cosine positive? What reference angle produces the cosine of? Then what angle satisfies the equation?
Eample 4: Find all angles that satisf the given trigonometric equation. 0, 360 0, 2 A. tan 3 B. sin 0 C. sin D. csc 2 E. cot 0 F. csc G. 2cos21 H. sin 2 sin I. sec 20 J. 2 tan 60 Think About It: Find the values of for the given trigonometric equation. cos.396 0, 360 For trigonometric equations which are difficult or impossible to solve using algebraic methods, ou can still use a calculator to solve them. Make sure to build a window which will show onl the solutions that ou want if solving b graphing. Eample 5: Use a calculator to find the solutions to 4cos in the interval,. Epress our answers to 3 decimal place accurac.
Stud Notes 5 Right Trig Identities (Section 8.6) Identities are true for all angles. Pthagorean Identities Reciprocal Identities Negative Identities cos 2 + sin 2 = 1 1 + tan 2 = sec 2 1 + cot 2 = csc 2 Ratio Identities tan cot sin cos cos sin Co function Identities csc 1 sin sec 1 cos cot 1 tan sin = sin( ) csc = csc( ) cos = cos( ) sec = sec( ) tan = tan( ) cot = cot( ) sin cos cos sin tan cot csc sec sec csc cot tan Prove the following identities: 1. sin seccsctan 2. tan cot sec csc 3. cos 4. tan REMEMBER!! Don t invent new rules. Changing things to sin and cos helps. You can t use Pthagorean unless things are squared. Don t move things across the equal sign when proving identities.