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Transcript
7.5 Apply the Tangent Ratio Goal Your Notes p Use the tangent ratio for indirect measurement. VOCABULARY Trigonometric ratio Tangent TANGENT RATIO Remember these abbreviations: tangent → tan opposite → opp. adjacent → adj. Let nABC be a right triangle with acute ∠A. The tangent of ∠A (written as tan A) is defined as follows: C leg opposite aA hypotenuse B leg adjacent to aA A length of leg opposite ∠A 5 tan A 5 }}} length of leg adjacent to ∠A Example 1 Find tangent ratios Find tan S and tan R. Write each answer as a fraction and as a decimal rounded to four places, if necessary. S 65 25 T Unless told otherwise, round values of trigonometric ratios to the ten-thousandths’ place and round lengths to the tenths’ place. 192 60 R Solution opp. ∠S adj. to ∠S tan S 5 } 5 opp. ∠R 5 tan R 5 } adj. to ∠R Lesson 7.5 • Geometry Notetaking Guide 5 5 5 5 5 < Copyright © Holt McDougal. All rights reserved. 7.5 Apply the Tangent Ratio Goal Your Notes p Use the tangent ratio for indirect measurement. VOCABULARY Trigonometric ratio A trigonometric ratio is a ratio of the lengths of two sides in a right triangle. Tangent The ratio of the length of the leg opposite an acute angle in a right triangle to the length of the leg adjacent to the angle is the tangent of the angle. TANGENT RATIO Remember these abbreviations: tangent → tan opposite → opp. adjacent → adj. Let nABC be a right triangle with acute ∠A. The tangent of ∠A (written as tan A) is defined as follows: C leg opposite aA B leg adjacent to aA A length of leg opposite ∠A BC 5 tan A 5 }}} length of leg adjacent to ∠A Example 1 hypotenuse AB Find tangent ratios Find tan S and tan R. Write each answer as a fraction and as a decimal rounded to four places, if necessary. S 65 25 T Unless told otherwise, round values of trigonometric ratios to the ten-thousandths’ place and round lengths to the tenths’ place. 192 60 R Solution opp. ∠S adj. to ∠S tan S 5 } 5 opp. ∠R 5 tan R 5 } adj. to ∠R Lesson 7.5 • Geometry Notetaking Guide RT 5 ST ST RT 60 5 25 5 25 60 12 5 2.4 5 5 5 < 0.4167 12 Copyright © Holt McDougal. All rights reserved. Your Notes Checkpoint Find tan B and tan C. Write each answer as a fraction and as a decimal rounded to four places. 1. 48 A 14 C 50 B Example 2 Find a leg length Find the value of x. Use the tangent of an acute angle to find a leg length. 17 318 x tan 318 5 Write ratio for tangent of 318. tan 318 5 Substitute. p tan 318 5 Example 3 Multiply each side by x5 Divide each side by x< Use a calculator to find . x< Simplify. . . Estimate height using tangent Lighthouse Find the height h of the lighthouse to the nearest foot. opp. p Copyright © Holt McDougal. All rights reserved. 5} adj. Write ratio for . 5 Substitute. 5h Multiply each side by <h Use a calculator and simplify. h 628 100 ft Lesson 7.5 • Geometry Notetaking Guide . 193 Your Notes Checkpoint Find tan B and tan C. Write each answer as a fraction and as a decimal rounded to four places. 1. 48 A 14 C 50 B 24 7 tan B 5 } < 3.4286, tan C 5 } < 0.2917 7 Example 2 24 Find a leg length Find the value of x. Use the tangent of an acute angle to find a leg length. 17 318 x opp. tan 318 5 } adj. Write ratio for tangent of 318. 17 tan 318 5 } Substitute. x x p tan 318 5 17 Multiply each side by x . 17 x5 } Divide each side by tan 318 . 17 x< } Use a calculator to find tan 318 . x < 28.3 Simplify. tan 318 0.6009 Example 3 Estimate height using tangent Lighthouse Find the height h of the lighthouse to the nearest foot. opp. tan 628 5 } adj. Write ratio for tan 628 . h tan 628 5 } Substitute. 100 h 628 100 ft 100 p tan 628 5 h Multiply each side by 100 . 188 < h Use a calculator and simplify. Copyright © Holt McDougal. All rights reserved. Lesson 7.5 • Geometry Notetaking Guide 193 Your Notes Use a special right triangle to find a tangent Example 4 Use a special right triangle to find the tangent of a 308 angle. Solution as the length of the shorter leg to Step 1 Choose simplify calculations. Use the 308-608-908 Triangle Theorem to find the length of the longer leg. 3 308 x longer leg 5 5 x5 Step 2 Find tan 308. The tangents of all 308 angles are the same constant ratio. Any right triangle with a 308 angle can be used to determine this value. tan 308 5 Write ratio for tangent of 308. tan 308 5 Substitute. The tangent of any 308 angle is < . Checkpoint In Exercises 2 and 3, find the value of x. Round to the nearest tenth. 2. x 3. 638 x 21 598 13 4. In Example 4, suppose the length of the shorter leg } is 1 instead of }Ï 3 . Show that the tangent of 308 is Homework Ï3 . still equal to } 3 194 Lesson 7.5 • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved. Your Notes Use a special right triangle to find a tangent Example 4 Use a special right triangle to find the tangent of a 308 angle. Solution } Step 1 Choose Ï3 as the length of the shorter leg to simplify calculations. Use the 308-608-908 Triangle Theorem to find the length of the longer leg. 3 308 x } longer leg 5 shorter leg p Ï3 } } x 5 Ï3 p Ï3 5 3 Step 2 Find tan 308. The tangents of all 308 angles are the same constant ratio. Any right triangle with a 308 angle can be used to determine this value. opp. adj. tan 308 5 } Write ratio for tangent of 308. } Ï3 tan 308 5 } 3 Substitute. } Ï3 The tangent of any 308 angle is } < 0.5774 . 3 Checkpoint In Exercises 2 and 3, find the value of x. Round to the nearest tenth. 2. x 3. 638 x 21 598 13 x < 6.6 x < 34.9 4. In Example 4, suppose the length of the shorter leg } is 1 instead of }Ï 3 . Show that the tangent of 308 is Homework Ï3 . still equal to } 3 } longer leg 5 shorter leg p Î3 } } x 5 1 p Ï3 5 Ï3 } } opp. Ï3 Ï3 1 1 p} tan 308 5 } 5 } } 5 } } } 5 } 3 adj. Ï3 Ï3 Ï3 194 Lesson 7.5 • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.