4.3. Right Triangle Trigonometry. The Six Trigonometric Functions. What you should learn. Why you should learn it

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1 0_00.qxd /7/05 :0 AM Page 0 Section. 0 Right Triangle Trigonometry What you should learn Evaluate trigonometric functions of acute angles. Use the fundamental trigonometric identities. Use a calculator to evaluate trigonometric functions. Use trigonometric functions to model and solve real-life problems. The Six Trigonometric Functions Our second look at the trigonometric functions is from a right triangle perspective. Consider a right triangle, with one acute angle labeled, as shown in Figure.6. Relative to the angle, the three sides of the triangle are the hypotenuse, the opposite side (the side opposite the angle ), and the adjacent side (the side adjacent to the angle ). Side opposite. Right Triangle Trigonometry us e Why you should learn it Hy po ten Trigonometric functions are often used to analyze real-life situations. For instance, in Exercise 7 on page, you can use trigonometric functions to find the height of a helium-filled balloon. Side adjacent to FIGURE.6 Using the lengths of these three sides, you can form six ratios that define the six trigonometric functions of the acute angle. sine cosecant cosine secant tangent cotangent In the following definitions, it is important to see that 0 < < 90 lies in the first quadrant) and that for such angles the value of each trigonometric function is positive. Right Triangle Definitions of Trigonometric Functions Let be an acute angle of a right triangle. The six trigonometric functions of the angle are defined as follows. (Note that the functions in the second row are the reciprocals of the corresponding functions in the first row.) sin opp hyp cos adj hyp tan opp adj csc hyp opp sec hyp adj cot adj opp Joseph Sohm; Chromosohm The abbreviations opp, adj, and hyp represent the lengths of the three sides of a right triangle. opp the length of the side opposite adj the length of the side adjacent to hyp the length of the hypotenuse

2 0_00.qxd /7/05 :0 AM Page 0 0 Chapter Trigonometry Example Evaluating Trigonometric Functions FIGURE.7 Hypotenuse You may wish to review the Pythagorean Theorem before presenting the examples in this section. Historical Note Georg Joachim Rhaeticus (5 576) was the leading Teutonic mathematical astronomer of the 6th century. He was the first to define the trigonometric functions as ratios of the sides of a right triangle. 5 FIGURE.8 5 Use the triangle in Figure.7 to find the values of the six trigonometric functions of. By the Pythagorean Theorem, hyp opp adj, it follows that hyp 5 5. So, the six trigonometric functions of sin opp hyp 5 cos adj hyp 5 tan opp adj Now try Exercise. In Example, you were given the lengths of two sides of the right triangle, but not the angle. Often, you will be asked to find the trigonometric functions of a given acute angle. To do this, construct a right triangle having as one of its angles. Example are Evaluating Trigonometric Functions of 5 Find the values of sin 5, cos 5, and tan 5. Construct a right triangle having 5 as one of its acute angles, as shown in Figure.8. Choose the length of the adjacent side to be. From geometry, you know that the other acute angle is also 5. So, the triangle is isosceles and the length of the opposite side is also. Using the Pythagorean Theorem, you find the length of the hypotenuse to be. sin 5 opp hyp cos 5 adj hyp tan 5 opp adj Now try Exercise 7. csc hyp opp 5 sec hyp adj 5 cot adj opp.

3 0_00.qxd /7/05 :0 AM Page 0 Section. Right Triangle Trigonometry 0 Example Evaluating Trigonometric Functions of 0 and 60 Because the angles 0, 5, and 60 6,, and occur frequently in trigonometry, you should learn to construct the triangles shown in Figures.8 and.9. Use the equilateral triangle shown in Figure.9 to find the values of sin 60, cos 60, sin 0, and cos 0. 0 Consider having your students construct the triangle in Figure.9 with angles in the corresponding radian measures, then find the six trigonometric functions for each of the acute angles. Technology You can use a calculator to convert the answers in Example to decimals. However, the radical form is the exact value and in most cases, the exact value is preferred. FIGURE.9 Use the Pythagorean Theorem and the equilateral triangle in Figure.9 to verify the lengths of the sides shown in the figure. For you have adj, opp, and hyp. So, sin 60 opp hyp 0, and For adj, opp, and hyp. So, sin 0 opp hyp and Now try Exercise , cos 60 adj hyp. cos 0 adj hyp. The triangles in Figures.7,.8, and.9 are useful problem-solving aids. Encourage your students to draw diagrams when they solve problems similar to those in Examples,, and. Sines, Cosines, and Tangents of Special Angles sin 0 sin cos 0 cos tan 0 tan sin 5 sin sin 60 sin cos 5 cos cos 60 cos tan 5 tan tan 60 tan In the box, note that sin 0 cos 60. This occurs because 0 and 60 are complementary angles. In general, it can be shown from the right triangle definitions that cofunctions of complementary angles are equal. That is, if is an acute angle, the following relationships are true. sin90 cos cos90 sin tan90 cot cot90 tan sec90 csc csc90 sec

4 0_00.qxd /7/05 :0 AM Page 0 0 Chapter Trigonometry Trigonometric Identities In trigonometry, a great deal of time is spent studying relationships between trigonometric functions (identities). These identities will be used many times in trigonometry and later in calculus. Encourage your students to learn them well. Fundamental Trigonometric Identities Reciprocal Identities sin csc csc sin Quotient Identities tan sin cos Pythagorean Identities sin cos cos sec sec cos cot cos sin tan sec cot csc tan cot cot tan Note that sin represents sin, cos represents cos, and so on. Example Applying Trigonometric Identities FIGURE Let be an acute angle such that sin 0.6. Find the values of (a) cos and (b) tan using trigonometric identities. a. To find the value of cos, use the Pythagorean identity sin cos. So, you have 0.6 cos Substitute 0.6 for sin. Subtract 0.6 from each side. Extract the positive square root. b. Now, knowing the sine and cosine of, you can find the tangent of to be tan sin cos cos cos Use the definitions of cos and tan, and the triangle shown in Figure.0, to check these results. Now try Exercise 9.

5 0_00.qxd /7/05 :0 AM Page 05 Section. Right Triangle Trigonometry 05 Example 5 Applying Trigonometric Identities Let be an acute angle such that tan. Find the values of (a) cot and (b) sec using trigonometric identities. a. cot Reciprocal identity tan 0 cot FIGURE. You can also use the reciprocal identities for sine, cosine, and tangent to evaluate the cosecant, secant, and cotangent functions with a calculator. For instance, you could use the following keystroke sequence to evaluate sec 8. COS 8 ENTER The calculator should display.570. One of the most common errors students make when they evaluate trigonometric functions with a calculator is not having their calculators set to the correct mode (radian vs. degree). b. sec tan Pythagorean identity sec sec 0 Use the definitions of cot and sec, and the triangle shown in Figure., to check these results. Now try Exercise. Evaluating Trigonometric Functions with a Calculator To use a calculator to evaluate trigonometric functions of angles measured in degrees, first set the calculator to degree mode and then proceed as demonstrated in Section.. For instance, you can find values of cos 8 and sec 8 as follows. Function Mode Calculator Keystrokes Display a. cos 8 Degree COS 8 ENTER b. sec 8 Degree COS 8 x ENTER.570 Throughout this text, angles are assumed to be measured in radians unless noted otherwise. For example, sin means the sine of radian and sin means the sine of degree. Using a Calculator Use a calculator to evaluate sec5 0. Begin by converting to decimal degree form. [Recall that sec 0 Example Then, use a calculator to evaluate sec Function Calculator Keystrokes Display sec5 0 sec 5.67 COS 5.67 x ENTER Now try Exercise 7. and

6 0_00.qxd /7/05 :0 AM Page Chapter Trigonometry Observer Observer FIGURE. y Object Angle of elevation Horizontal Horizontal Angle of depression Object Angle of elevation 78. Applications Involving Right Triangles Many applications of trigonometry involve a process called solving right triangles. In this type of application, you are usually given one side of a right triangle and one of the acute angles and are asked to find one of the other sides, or you are given two sides and are asked to find one of the acute angles. In Example 7, the angle you are given is the angle of elevation, which represents the angle from the horizontal upward to an object. For objects that lie below the horizontal, it is common to use the term angle of depression, as shown in Figure.. Example 7 Using Trigonometry to Solve a Right Triangle A surveyor is standing 5 feet from the base of the Washington Monument, as shown in Figure.. The surveyor measures the angle of elevation to the top of the monument as 78.. How tall is the Washington Monument? From Figure., you can see that tan 78. opp adj y x where x 5 and y is the height of the monument. So, the height of the Washington Monument is y x tan feet. Now try Exercise 6. FIGURE. x = 5 ft Not drawn to scale Example 8 Using Trigonometry to Solve a Right Triangle An historic lighthouse is 00 yards from a bike path along the edge of a lake. A walkway to the lighthouse is 00 yards long. Find the acute angle between the bike path and the walkway, as illustrated in Figure.. 00 yd 00 yd FIGURE. From Figure., you can see that the sine of the angle sin opp 00 hyp Now you should recognize that Now try Exercise 65. is

7 0_00.qxd /7/05 :0 AM Page 07 Section. Right Triangle Trigonometry 07 0 By now you are able to recognize that is the acute angle that satisfies the equation sin. Suppose, however, that you were given the equation sin 0.6 and were asked to find the acute angle. Because sin 0 and sin you might guess that lies somewhere between 0 and 5. In a later section, you will study a method by which a more precise value of can be determined. Example 9 Solving a Right Triangle Find the length c of the skateboard ramp shown in Figure c ft Activities. Use the right triangle shown to find each of the six trigonometric functions of the angle. Answer: sin 9 9, csc 9, cos 59 9, sec 9 5, 5 tan 5, cot 5. A 0-foot ladder leans against the side of a house. The ladder makes an angle of 60 with the ground. How far up the side of the house does the ladder reach? Answer: feet FIGURE.5 From Figure.5, you can see that sin 8. opp hyp So, the length of the skateboard ramp is c c. sin feet. Now try Exercise 67.

8 0_00.qxd /7/05 :0 AM Page Chapter Trigonometry. Exercises VOCABULARY CHECK:. Match the trigonometric function with its right triangle definition. (a) Sine (b) Cosine (c) Tangent (d) Cosecant (e) Secant (f) Cotangent hypotenuse adjacent hypotenuse adjacent opposite opposite (i) (ii) (iii) (iv) (v) (vi) adjacent opposite opposite hypotenuse hypotenuse adjacent In Exercises and, fill in the blanks.. Relative to the angle, the three sides of a right triangle are the side, the side, and the.. An angle that measures from the horizontal upward to an object is called the angle of, whereas an angle that measures from the horizontal downward to an object is called the angle of. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at In Exercises, find the exact values of the six trigonometric functions of the angle shown in the figure. (Use the Pythagorean Theorem to find the third side of the triangle.) In Exercises 5 8, find the exact values of the six trigonometric functions of the angle for each of the two triangles. Explain why the function values are the same In Exercises 9 6, sketch a right triangle corresponding to the trigonometric function of the acute angle. Use the Pythagorean Theorem to determine the third side and then find the other five trigonometric functions of. 9. sin 0. cos 5 7. sec. cot 5. tan. sec 6 5. cot 6. csc 7 In Exercises 7 6, construct an appropriate triangle to complete the table. 0 90, 0 / Function (deg) (rad) Function Value 7. sin 0 8. cos 5 9. tan 0. sec. cot. csc. cos. sin 5. cot 6. tan 6

9 0_00.qxd /7/05 :0 AM Page 09 Section. Right Triangle Trigonometry 09 In Exercises 7, use the given function value(s), and trigonometric identities (including the cofunction identities), to find the indicated trigonometric functions. 7. sin 60 cos 60, (a) tan 60 (b) sin 0 (c) cos 0 (d) cot sin 0 tan 0, (a) csc 0 (b) cot 60 (c) cos 0 (d) cot 0 9. (a) sin (b) cos (c) tan (d) sec90 0. sec 5, tan 6 (a) cos (b) cot (c) cot90 (d) sin. cos (a) sec (b) sin (c) cot (d) sin90. tan 5 (a) cot (b) cos (c) tan90 (d) csc In Exercises, use trigonometric identities to transform the left side of the equation into the right side 0 < < /.. tan cot. cos sec 5. tan cos sin 6. cot sin cos 7. cos cos sin 8. sin sin cos 9. sec tan sec tan 0. sin cos sin.. csc, sec sin cos csc sec cos sin tan cot csc tan In Exercises 5, use a calculator to evaluate each function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.). (a) sin 0 (b) cos 80. (a) tan.5 (b) cot (a) sin 6.5 (b) csc (a) cos 6 8 (b) sin (a) sec (b) csc (a) cos 50 5 (b) sec (a) cot 5 (b) tan (a) sec (b) cos (a) csc 0 (b) tan (a) sec (b) cot In Exercises 5 58, find the values of in degrees 0 < < 90 and radians 0 < < / without the aid of a calculator. 5. (a) sin (b) csc 5. (a) cos (b) tan 55. (a) sec (b) cot 56. (a) tan (b) cos 57. (a) csc (b) sin 58. (a) cot (b) sec In Exercises 59 6, solve for x, y, or r as indicated. 59. Solve for x. 60. Solve for y Solve for x. 6. Solve for r. 60 x x 0 8 y Empire State Building You are standing 5 meters from the base of the Empire State Building. You estimate that the angle of elevation to the top of the 86th floor (the observatory) is 8. If the total height of the building is another meters above the 86th floor, what is the approximate height of the building? One of your friends is on the 86th floor. What is the distance between you and your friend? r 0

10 0_00.qxd /7/05 :0 AM Page 0 0 Chapter Trigonometry 6. Height A six-foot person walks from the base of a broadcasting tower directly toward the tip of the shadow cast by the tower. When the person is feet from the tower and feet from the tip of the shadow, the person s shadow starts to appear beyond the tower s shadow. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities of the triangle and use a variable to indicate the height of the tower. (b) Use a trigonometric function to write an equation involving the unknown quantity. (c) What is the height of the tower? 65. Angle of Elevation You are skiing down a mountain with a vertical height of 500 feet. The distance from the top of the mountain to the base is 000 feet. What is the angle of elevation from the base to the top of the mountain? 66. Width of a River A biologist wants to know the width w of a river so in order to properly set instruments for studying the pollutants in the water. From point A, the biologist walks downstream 00 feet and sights to point C (see figure). From this sighting, it is determined that How wide is the river? 5. w C = 5 A 00 ft 67. Length A steel cable zip-line is being constructed for a competition on a reality television show. One end of the zip-line is attached to a platform on top of a 50-foot pole. The other end of the zip-line is attached to the top of a 5-foot stake. The angle of elevation to the platform is (see figure). 68. Height of a Mountain In traveling across flat land, you notice a mountain directly in front of you. Its angle of elevation (to the peak) is.5. After you drive miles closer to the mountain, the angle of elevation is 9. Approximate the height of the mountain. 69. Machine Shop Calculations A steel plate has the form of one-fourth of a circle with a radius of 60 centimeters. Two two-centimeter holes are to be drilled in the plate positioned as shown in the figure. Find the coordinates of the center of each hole y.5 9 mi ( x, y ) ( x, y ) Machine Shop Calculations A tapered shaft has a diameter of 5 centimeters at the small end and is 5 centimeters long (see figure). The taper is. Find the diameter d of the large end of the shaft. Not drawn to scale x 5 cm d 5 ft = 50 ft 5 cm (a) How long is the zip-line? (b) How far is the stake from the pole? (c) Contestants take an average of 6 seconds to reach the ground from the top of the zip-line. At what rate are contestants moving down the line? At what rate are they dropping vertically?

11 0_00.qxd /7/05 :0 AM Page Section. Right Triangle Trigonometry 7. Height A 0-meter line is used to tether a heliumfilled balloon. Because of a breeze, the line makes an angle of approximately 85 with the ground. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities of the triangle and use a variable to indicate the height of the balloon. (b) Use a trigonometric function to write an equation involving the unknown quantity. (c) What is the height of the balloon? (d) The breeze becomes stronger and the angle the balloon makes with the ground decreases. How does this affect the triangle you drew in part (a)? (e) Complete the table, which shows the heights (in meters) of the balloon for decreasing angle measures. Angle, Height Angle, Height Model It 7. Geometry Use a compass to sketch a quarter of a circle of radius 0 centimeters. Using a protractor, construct an angle of 0 in standard position (see figure). Drop a perpendicular line from the point of intersection of the terminal side of the angle and the arc of the circle. By actual measurement, calculate the coordinates x, y of the point of intersection and use these measurements to approximate the six trigonometric functions of a 0 angle. y (f) As the angle the balloon makes with the ground approaches 0, how does this affect the height of the balloon? Draw a right triangle to explain your reasoning. Synthesis True or False? In Exercises 7 78, determine whether the statement is true or false. Justify your answer. 7. sin 60 csc sec 0 csc sin 5 cos cot 0 csc 0 sin sin 78. tan5 tan 5 sin Writing In right triangle trigonometry, explain why sin 0 regardless of the size of the triangle. 80. Think About It You are given only the value tan. Is it possible to find the value of sec without finding the measure of? Explain. 8. Exploration (a) Complete the table. (b) Is or sin greater for in the interval 0, 0.5? (c) As approaches 0, how do and sin compare? Explain. 8. Exploration (a) Complete the table. sin cos sin (b) Discuss the behavior of the sine function for in the range from 0 to 90. (c) Discuss the behavior of the cosine function for in the range from 0 to 90. (d) Use the definitions of the sine and cosine functions to explain the results of parts (b) and (c). Skills Review In Exercises 8 86, perform the operations and simplify. 0 cm 0 (x, y) 0 x x 6x x x x x 6 x 6 t 5t t 6 9 t t t x x x x x x x