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- 1. 13-1 Right-Angle Trigonometry Holt Algebra 2 Warm Up Lesson Presentation Lesson Quiz
- 2. Warm Up Given the measure of one of the acute angles in a right triangle, find the measure of the other acute angle. 1. 45° 2. 60° 3. 24° 4. 38° 45° 30° 66° 52°
- 3. Warm Up Continued Find the unknown length for each right triangle with legs a and b and hypotenuse c . 5. b = 12, c =13 6. a = 3, b = 3 a = 5
- 4. Understand and use trigonometric relationships of acute angles in triangles. Determine side lengths of right triangles by using trigonometric functions. Objectives
- 5. trigonometric function sine cosine tangent cosecants secant cotangent Vocabulary
- 6. A trigonometric function is a function whose rule is given by a trigonometric ratio. A trigonometric ratio compares the lengths of two sides of a right triangle. The Greek letter theta θ is traditionally used to represent the measure of an acute angle in a right triangle. The values of trigonometric ratios depend upon θ .
- 7. The triangle shown at right is similar to the one in the table because their corresponding angles are congruent. No matter which triangle is used, the value of sin θ is the same. The values of the sine and other trigonometric functions depend only on angle θ and not on the size of the triangle.
- 8. Example 1: Finding Trigonometric Ratios Find the value of the sine, cosine, and tangent functions for θ . sin θ = cos θ = tan θ =
- 9. Check It Out! Example 1 Find the value of the sine, cosine, and tangent functions for θ . sin θ = cos θ = tan θ =
- 10. You will frequently need to determine the value of trigonometric ratios for 30°,60°, and 45° angles as you solve trigonometry problems. Recall from geometry that in a 30°-60°-90° triangle, the ration of the side lengths is 1: 3 :2, and that in a 45°-45°-90° triangle, the ratio of the side lengths is 1:1: 2.
- 11. Example 2: Finding Side Lengths of Special Right Triangles Use a trigonometric function to find the value of x . x = 37 The sine function relates the opposite leg and the hypotenuse. Multiply both sides by 74 to solve for x. ° Substitute for sin 30°. Substitute 30° for θ , x for opp, and 74 for hyp.
- 12. Check It Out! Example 2 Use a trigonometric function to find the value of x . The sine function relates the opposite leg and the hypotenuse. Multiply both sides by 20 to solve for x. Substitute 45 for θ , x for opp, and 20 for hyp. ° ° Substitute for sin 45°.
- 13. Example 3: Sports Application In a waterskiing competition, a jump ramp has the measurements shown. To the nearest foot, what is the height h above water that a skier leaves the ramp? 5 ≈ h The height above the water is about 5 ft. Substitute 15.1° for θ , h for opp., and 19 for hyp. Multiply both sides by 19. Use a calculator to simplify.
- 14. Make sure that your graphing calculator is set to interpret angle values as degrees. Press . Check that Degree and not Radian is highlighted in the third row. Caution!
- 15. Check It Out! Example 3 A skateboard ramp will have a height of 12 in., and the angle between the ramp and the ground will be 17°. To the nearest inch, what will be the length l of the ramp? l ≈ 41 The length of the ramp is about 41 in. Substitute 17° for θ , l for hyp., and 12 for opp. Multiply both sides by l and divide by sin 17°. Use a calculator to simplify.
- 16. When an object is above or below another object, you can find distances indirectly by using the angle of elevation or the angle of depression between the objects.
- 17. Example 4: Geology Application A biologist whose eye level is 6 ft above the ground measures the angle of elevation to the top of a tree to be 38.7°. If the biologist is standing 180 ft from the tree’s base, what is the height of the tree to the nearest foot? Step 1 Draw and label a diagram to represent the information given in the problem.
- 18. Example 4 Continued Step 2 Let x represent the height of the tree compared with the biologist’s eye level. Determine the value of x . Use the tangent function. 180 (tan 38.7°) = x Substitute 38.7 for θ , x for opp., and 180 for adj. Multiply both sides by 180. 144 ≈ x Use a calculator to solve for x.
- 19. Example 4 Continued Step 3 Determine the overall height of the tree. x + 6 = 144 + 6 = 150 The height of the tree is about 150 ft.
- 20. Check It Out! Example 4 A surveyor whose eye level is 6 ft above the ground measures the angle of elevation to the top of the highest hill on a roller coaster to be 60.7°. If the surveyor is standing 120 ft from the hill’s base, what is the height of the hill to the nearest foot? Step 1 Draw and label a diagram to represent the information given in the problem. 120 ft 60.7°
- 21. Check It Out! Example 4 Continued Use the tangent function. 120 (tan 60.7°) = x Substitute 60.7 for θ , x for opp., and 120 for adj. Multiply both sides by 120. Step 2 Let x represent the height of the hill compared with the surveyor’s eye level. Determine the value of x . 214 ≈ x Use a calculator to solve for x.
- 22. Check It Out! Example 4 Continued Step 3 Determine the overall height of the roller coaster hill. x + 6 = 214 + 6 = 220 The height of the hill is about 220 ft.
- 23. The reciprocals of the sine, cosine, and tangent ratios are also trigonometric ratios. They are trigonometric functions, cosecant, secant, and cotangent.
- 24. Example 5: Finding All Trigonometric Functions Find the values of the six trigonometric functions for θ . Step 1 Find the length of the hypotenuse. a 2 + b 2 = c 2 c 2 = 24 2 + 70 2 c 2 = 5476 c = 74 Pythagorean Theorem. Substitute 24 for a and 70 for b. Simplify. Solve for c. Eliminate the negative solution. 70 24 θ
- 25. Example 5 Continued Step 2 Find the function values.
- 26. In each reciprocal pair of trigonometric functions, there is exactly one “ co ” Helpful Hint
- 27. Find the values of the six trigonometric functions for θ . Step 1 Find the length of the hypotenuse. a 2 + b 2 = c 2 c 2 = 18 2 + 80 2 c 2 = 6724 c = 82 Pythagorean Theorem. Substitute 18 for a and 80 for b. Simplify. Solve for c. Eliminate the negative solution. Check It Out! Example 5 80 18 θ
- 28. Check It Out! Example 5 Continued Step 2 Find the function values.
- 29. Lesson Quiz: Part I Solve each equation. Check your answer. 1. Find the values of the six trigonometric functions for θ .
- 30. Lesson Quiz: Part II 2. Use a trigonometric function to find the value of x . 3. A helicopter’s altitude is 4500 ft, and a plane’s altitude is 12,000 ft. If the angle of depression from the plane to the helicopter is 27.6°, what is the distance between the two, to the nearest hundred feet? 16,200 ft

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