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## Right-Angle Trigonometry

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**Right-Angle Trigonometry**Essential Questions • How do we understand and use trigonometric relationships of acute angles in triangles? • How do we determine side lengths of right triangles by using trigonometric functions? Holt McDougal Algebra 2 Holt Algebra 2**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.**Finding Trigonometric Ratios**Find the value of the sine, cosine, and tangent functions for θ. hyp. opp. adj. sin θ = cos θ = tan θ =**Finding Trigonometric Ratios**Find the value of the sine, cosine, and tangent functions for θ. hyp. adj. opp. sin θ = cos θ = tan θ =**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.**Substitute 30° for θ, x for opp, and 74 for hyp.**° Substitute for sin 30°. Finding Side Lengths of Special Right Triangles Use a trigonometric function to find the value of x. Which function relates the opposite and the hypotenuse? hyp. opp. 2x = 74 Solve for x. x = 37**°**° Substitute 45 for θ, x for opp, and 20 for hyp. Finding Side Lengths of Special Right Triangles Use a trigonometric function to find the value of x. Which function relates the opposite and the hypotenuse? hyp. opp. Substitute for sin 45°. Solve for x.**Caution!**Make sure that your calculator is set to interpret angle values as degrees. Press DRG. Check that Degree and not Radian is highlighted.**Finding Side Lengths of Other Right Triangles**Use a trigonometric function to find the value of x. Which function relates the opposite and the adjacent? Substitute 38 for θ, 42 for opp., and x for adj. opp. Because x is on bottom, divide. adj.**Finding Side Lengths of Other Right Triangles**Use a trigonometric function to find the value of x. hyp. Which function relates the adjacent and the hypotenuse? adj. Substitute 52 for θ, x for adj., and 155 for hyp. Because x is on top, multiply.