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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° 45° 2. 60° 30° 3. 24° 66° 4. 38° 52° 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 Objectives Understand and use trigonometric relationships of acute angles in triangles. Determine side lengths of right triangles by using trigonometric functions. Vocabulary trigonometric function sine cosine tangent cosecants secant cotangent 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. 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. Example 1: Finding Trigonometric Ratios Find the value of the sine, cosine, and tangent functions for θ. sin θ = cos θ = tan θ = Example 2 Find the value of the sine, cosine, and tangent functions for θ. sin θ = cos θ = tan θ = 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? Substitute 15.1° for θ, h for opp., and 19 for hyp. Multiply both sides by 19. Use a calculator to simplify. 5≈h The height above the water is about 5 ft. Caution! 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. Example 4 You Try! 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? 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. l ≈ 41 The length of the ramp is about 41 in. The reciprocals of the sine, cosine, and tangent ratios are also trigonometric ratios. They are trigonometric functions, cosecant, secant, and cotangent. 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 = c2 c2 = 242 + 702 c2 = 5476 c = 74 Pythagorean Theorem. Substitute 24 for a and 70 for b. Simplify. Solve for c. Eliminate the negative solution. 70 θ 24 Example 5 Continued Step 2 Find the function values. Helpful Hint In each reciprocal pair of trigonometric functions, there is exactly one “co” Example 6 You Try! Find the values of the six trigonometric functions for θ. Step 1 Find the length of the hypotenuse. a2 + b2 = c2 Pythagorean Theorem. c2 = 182 + 802 Substitute 18 for a and 80 for b. c2 = 6724 Simplify. c = 82 Solve for c. Eliminate the negative solution. 80 θ 18 Example 6 Continued Step 2 Find the function values. Lesson Quiz: Part I Solve each equation. Check your answer. 1. Find the values of the six trigonometric functions for θ.