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7.6 Apply the Sine and Cosine Ratios Goal Your Notes p Use the sine and cosine ratios. VOCABULARY Sine, cosine Angle of elevation Angle of depression SINE AND COSINE RATIOS C Let nABC be a right triangle leg with acute ∠A. The sine of ∠A opposite aA and cosine of ∠A (written sin A B and cos A) are defined as follows: Remember these abbreviations: sine → sin cosine → cos hypotenuse → hyp hypotenuse leg adjacent to aA A length of leg opposite ∠A 5 sin A 5 }}} length of hypotenuse length of leg adjacent to ∠A 5 cos A 5 }}} length of hypotenuse Copyright © Holt McDougal. All rights reserved. Lesson 7.6 • Geometry Notetaking Guide 195 7.6 Apply the Sine and Cosine Ratios Goal Your Notes p Use the sine and cosine ratios. VOCABULARY Sine, cosine Sine and cosine are trigonometric ratios for acute angles that involve the lengths of a leg and the hypotenuse of a right triangle. Angle of elevation When looking up at an object, the angle your line of sight makes with a horizontal line is called the angle of elevation. Angle of depression When looking down at an object, the angle your line of sight makes with a horizontal line is called the angle of depression. SINE AND COSINE RATIOS C Let nABC be a right triangle leg with acute ∠A. The sine of ∠A opposite aA and cosine of ∠A (written sin A B and cos A) are defined as follows: Remember these abbreviations: sine → sin cosine → cos hypotenuse → hyp length of leg opposite ∠A 5 sin A 5 }}} length of hypotenuse length of leg adjacent to ∠A leg adjacent to aA A BC AC 5 cos A 5 }}} length of hypotenuse Copyright © Holt McDougal. All rights reserved. hypotenuse AB AC Lesson 7.6 • Geometry Notetaking Guide 195 Your Notes Find sine ratios Example 1 Find sin U and sin W. Write each answer as a fraction and as a decimal rounded to four places. U 34 30 Solution opp. ∠U 5 sin U 5 } hyp. W 5 opp. ∠W sin W 5 } 5 hyp. 5 5 5 16 V < < Find cosine ratios Example 2 Find cos S and cos R. Write each answer as a fraction and as a decimal rounded to four places. S 53 45 Solution adj. to ∠S cos S 5 } hyp. adj. to ∠R cos R 5 } hyp. R 5 < 5 < 28 T Checkpoint Find sin B, sin C, cos B, and cos C. Write each answer as a fraction and as a decimal rounded to four places. 1. A 20 21 C 29 B 196 Lesson 7.6 • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved. Your Notes Find sine ratios Example 1 Find sin U and sin W. Write each answer as a fraction and as a decimal rounded to four places. U 34 30 Solution opp. ∠U 5 sin U 5 } hyp. opp. ∠W sin W 5 } 5 hyp. WV UW 5 UV 5 UW 16 34 30 34 5 5 8 17 W 16 V < 0.4706 15 17 < 0.8824 Find cosine ratios Example 2 Find cos S and cos R. Write each answer as a fraction and as a decimal rounded to four places. S 53 45 Solution adj. to ∠S cos S 5 } hyp. adj. to ∠R cos R 5 } hyp. ST SR RT SR 5 5 45 53 28 53 R 28 T < 0.8491 < 0.5283 Checkpoint Find sin B, sin C, cos B, and cos C. Write each answer as a fraction and as a decimal rounded to four places. 1. A 20 21 C 29 B 20 21 < 0.6897, < 0.7241, sin C 5 } sin B 5 } 29 29 20 21 < 0.7241 < 0.6897, cos C 5 } cos B 5 } 29 29 196 Lesson 7.6 • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved. Your Notes Example 3 Use a trigonometric ratio to find a hypotenuse Basketball You walk from one corner of a basketball court to the opposite corner. Write and solve a proportion using a trigonometric ratio to approximate the distance of the walk. x ft 628 Solution sin 628 5 Write ratio for sine of 628. sin 628 5 Substitute. p 5 Multiply each side by Example 4 Divide each side by x< Use a calculator to find . x< Simplify. . feet. Find a hypotenuse using an angle of depression Roller Coaster You are at the top of a roller coaster 100 feet above the ground. The angle of depression is 448. About how far do you ride down the hill? 448 x ft 100 ft sin 448 5 Write ratio for sine of 448. sin 448 5 Substitute. 5 Multiply each side by x5 Divide each side by x< Use a calculator to find . x< Simplify. You ride about Copyright © Holt McDougal. All rights reserved. . x5 The distance of the walk is about xp 94 ft . . feet down the hill. Lesson 7.6 • Geometry Notetaking Guide 197 Your Notes Example 3 Use a trigonometric ratio to find a hypotenuse Basketball You walk from one corner of a basketball court to the opposite corner. Write and solve a proportion using a trigonometric ratio to approximate the distance of the walk. x ft 94 ft 628 Solution opp. hyp. Write ratio for sine of 628. 94 x Substitute. sin 628 5 } sin 628 5 } x p sin 628 5 94 Multiply each side by x . 94 sin 628 Divide each side by sin 628 . x< } 94 0.8829 Use a calculator to find sin 628 . x < 106.5 Simplify. x5 } The distance of the walk is about 106.5 feet. Example 4 Find a hypotenuse using an angle of depression Roller Coaster You are at the top of a roller coaster 100 feet above the ground. The angle of depression is 448. About how far do you ride down the hill? opp. hyp. sin 448 5 } 100 sin 448 5 } x x p sin 448 5 100 100 sin 448 x5 } 100 0.6947 448 x ft 100 ft Write ratio for sine of 448. Substitute. Multiply each side by x . Divide each side by sin 448 . x< } Use a calculator to find sin 448 . x < 143.9 Simplify. You ride about 144 feet down the hill. Copyright © Holt McDougal. All rights reserved. Lesson 7.6 • Geometry Notetaking Guide 197 Your Notes Checkpoint Complete the following exercises. 2. In Example 3, use the cosine ratio to approximate the width of the basketball court. 3. Suppose the angle of depression in Example 4 is 728. About how far would you ride down the hill? Example 5 Find leg lengths using an angle of elevation Railroad A railroad crossing arm that is 20 feet long is stuck with an angle of elevation of 358. Find the lengths x and y. 20 ft x ft 358 y ft Solution Step 1 Find x. opp. 5} hyp. Write ratio for of . 5 Substitute. 5x Multiply each side by <x Use a calculator to simplify. . Step 2 Find y. adj. 198 Lesson 7.6 • Geometry Notetaking Guide 5} hyp. Write ratio for of . 5 Substitute. 5y Multiply each side by <y Use a calculator to simplify. . Copyright © Holt McDougal. All rights reserved. Your Notes Checkpoint Complete the following exercises. 2. In Example 3, use the cosine ratio to approximate the width of the basketball court. about 50 feet 3. Suppose the angle of depression in Example 4 is 728. About how far would you ride down the hill? about 105 feet Example 5 Find leg lengths using an angle of elevation Railroad A railroad crossing arm that is 20 feet long is stuck with an angle of elevation of 358. Find the lengths x and y. 20 ft x ft 358 y ft Solution Step 1 Find x. opp. sin 358 5 } hyp. Write ratio for of 358 . x sin 358 5 } Substitute. 20 sine 20 p sin 358 5 x Multiply each side by 20 . 11.5 < x Use a calculator to simplify. Step 2 Find y. adj. cos 358 5 } hyp. y 20 cos 358 5 } 198 Lesson 7.6 • Geometry Notetaking Guide Write ratio for cosine of 358 . Substitute. 20 p cos 358 5 y Multiply each side by 20 . 16.4 < y Use a calculator to simplify. Copyright © Holt McDougal. All rights reserved. Your Notes Example 6 Use a special right triangle to find a sin and cos Use a special right triangle to find the sine and cosine of a 308 angle. Solution Use the 308-608-908 Triangle Theorem to draw a right } . Then set up triangle with side lengths of 1, Ï3 , and sine and cosine ratios for the 308 angle. sin 308 5 5 cos 308 5 5 3 5 308 < 1 2 608 Checkpoint Complete the following exercises. 4. In Example 5, suppose the angle of elevation is 408. What are the new lengths x and y ? 5. Use a special right triangle to find the sine and cosine of a 608 angle. Homework Copyright © Holt McDougal. All rights reserved. Lesson 7.6 • Geometry Notetaking Guide 199 Your Notes Example 6 Use a special right triangle to find a sin and cos Use a special right triangle to find the sine and cosine of a 308 angle. Solution Use the 308-608-908 Triangle Theorem to draw a right } triangle with side lengths of 1, Ï3 , and 2 . Then set up sine and cosine ratios for the 308 angle. opp. 1 sin 308 5 } 5 } 5 0.5000 hyp. 3 2 } adj. Ï3 cos 308 5 } 5 } < 0.8660 2 hyp. 308 1 2 608 Checkpoint Complete the following exercises. 4. In Example 5, suppose the angle of elevation is 408. What are the new lengths x and y ? x < 12.9, y < 15.3 5. Use a special right triangle to find the sine and cosine of a 608 angle. sin 608 < 0.8660 cos 608 5 0.5000 Homework Copyright © Holt McDougal. All rights reserved. Lesson 7.6 • Geometry Notetaking Guide 199 Focus On Trig Use after Lesson 7.6 Your Notes Cotangent, Secant, and Cosecant Ratios Goal p Use the cotangent, secant, and cosecant ratios. VOCABULARY Cotangent, secant, cosecant PROPERTY SUMMARY BOX Let nABC be a right triangle with acute /A. The cotangent of /A, secant of /A, and cosecant of /A (written as cot A, sec A, and csc A) are: /A length of leg cot A 5 }}} /A length of leg length of sec A 5 }}} /A length of leg Remember these abbreviations: cot cotangent sec secant csc cosecant length of csc A 5 }}} /A length of leg Note that these ratios are the reciprocals of the tangent, cosine, and sine ratios. 1 cot A 5 } Example 1 5} 5} 5} B leg opposite A C hypotenuse A leg adjacent to A 1 1 sec A 5 } csc A 5 } Find sec F. Find sec F. F hyp. sec F 5 } 5 } 5 adj. to /F 26 10 D 200 7.6 Focus on Trigonometry • Geometry Notetaking Guide 24 E Copyright © Holt McDougal. All rights reserved. Focus On Trig Use after Lesson 7.6 Your Notes Cotangent, Secant, and Cosecant Ratios Goal p Use the cotangent, secant, and cosecant ratios. VOCABULARY Cotangent, secant, cosecant Cotangent, secant, and cosecant are trigonometric ratios for acute angles involving the side lengths of a right triangle. PROPERTY SUMMARY BOX Let nABC be a right triangle with acute /A. The cotangent of /A, secant of /A, and cosecant of /A (written as cot A, sec A, and csc A) are: AC length of leg adjacent to /A }}} 5} cot A 5 BC opposite /A length of leg Remember these abbreviations: cot cotangent sec secant csc cosecant length of hypotenuse sec A 5 }}} length of leg adjacent to /A AB 5} AC length of hypotenuse csc A 5 }}} opposite /A length of leg AB 5} BC Note that these ratios are the reciprocals of the tangent, cosine, and sine ratios. 1 cot A 5 } tan A Example 1 B leg opposite A C hypotenuse A leg adjacent to A 1 1 sec A 5 } csc A 5 } cos A sin A Find sec F. Find sec F. F 26 hyp. sec F 5 } 5 } 5 2.6 adj. to /F 10 200 7.6 Focus on Trigonometry • Geometry Notetaking Guide 26 10 D 24 E Copyright © Holt McDougal. All rights reserved. Your Notes Checkpoint Find cot O and csc O. 1. F 17 8 O N 15 Use trigonometric ratios Example 2 Find the value of the variable. a. b. x 7 4.2 15° 63° l Solution If your calculator does not have a csc key, you can find sin 158 and use the x21 key to find csc 158. You are given an angle, a side it, and asked to find the length of the hypotenuse. You are given an angle, a side it, and asked to find the length of the side to it. Use cosecant 5 }. Use cotangent 5 }. 5 }. csc 5 }. cot csc 158 5 } cot 638 5 } 5x 5I ≈x ≈I Checkpoint Find the value of the variables. Homework 2. 36° n 29 m Copyright © Holt McDougal. All rights reserved. 7.6 Focus on Trigonometry • Geometry Notetaking Guide 201 Your Notes Checkpoint Find cot O and csc O. 1. 15 cot O 5 } 5 1.875, F 8 17 csc O 5 } = 2.125 8 17 8 O N 15 Use trigonometric ratios Example 2 Find the value of the variable. a. b. x 7 4.2 15° 63° l Solution You are given an angle, a side opposite it, and asked to find the length of the hypotenuse. hyp. } Use cosecant 5 . opp. If your calculator does not have a csc key, you can find sin 158 and use the x21 key to find csc 158. hyp. csc 158 5 }. opp. x You are given an angle, a side opposite it, and asked to find the length of the side adjacent to it. adj. } Use cotangent 5 . opp. adj. cot 638 5 }. opp. l csc 158 5 } 7 cot 638 5 } 7 csc 158 5 x 4.2 cot 638 5 I 27.05 ≈ x 2.14 ≈ I 4.2 Checkpoint Find the value of the variables. Homework 2. 36° n 29 hyp. adj. cot 5 } opp. n sec 368 5 } 29 sec 368 = n cot 368 5 29 } m 29 m5 } 35.85 ≈ n 21.07 ≈ m 29 m Copyright © Holt McDougal. All rights reserved. adj. sec 5 } cot 368 7.6 Focus on Trigonometry • Geometry Notetaking Guide 201