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Right-Angle Right-AngleTrigonometry Trigonometry • 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? HoltMcDougal Algebra 2Algebra 2 Holt Right-Angle Trigonometry The reciprocals of the sine, cosine, and tangent ratios are also trigonometric ratios. They are trigonometric functions, cosecant, secant, and cotangent. Holt McDougal Algebra 2 Right-Angle Trigonometry Finding All Trigonometric Functions 1. Find the values of the six trigonometric functions for θ. Find the length of the hypotenuse. a2 + b2 = 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. hyp. 74 70 opp. θ Find the lengths of the 6 trigonometric values. adj. 24 opp 70 35 24 12 adj opp 70 35 sin cos tan hyp hyp adj 74 37 74 37 24 12 1 37 csc sin 35 Holt McDougal Algebra 2 37 1 sec cos 12 cot 12 1 tan 35 Right-Angle Trigonometry Helpful Hint In each reciprocal pair of trigonometric functions, there is exactly one “co” Holt McDougal Algebra 2 Right-Angle Trigonometry Finding All Trigonometric Functions 2. Find the values of the six trigonometric functions for θ. Find the length of the hypotenuse. a2 + b2 = c2 c2 = 182 + 802 c2 =6724 c = 82 Pythagorean Theorem. Substitute 18 for a and 80 for b. Simplify. Solve for c. Eliminate the negative solution. hyp. 82 80 opp. θ Find the lengths of the 6 trigonometric values. adj. 18 opp 80 40 9 adj 18 opp 80 40 sin cos tan hyp 82 hyp 82 41 adj 18 41 9 1 41 csc sin 40 Holt McDougal Algebra 2 41 1 sec cos 9 cot 9 1 tan 40 Right-Angle Trigonometry Sports Application 3. In a waterskiing competition, a jump ramp has the measurements shown. To the nearest foot, what opp. is the height h above water that a skier leaves the ramp? hyp. Substitute 15.1° for θ, h for opp., and 19 for hyp. Multiply both sides by 19. 5≈h Use a calculator to simplify. The height above the water is about 5 ft. Holt McDougal Algebra 2 Right-Angle Trigonometry Sports Application 4. 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? hyp. opp. Substitute 17° for θ, l for hyp., and 12 for opp. Divide 12 by sine 17. l ≈ 41 Use a calculator to simplify. The length of the ramp is about 41 in. Holt McDougal Algebra 2 Right-Angle Trigonometry 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. Holt McDougal Algebra 2 Right-Angle Trigonometry Geology Application 5. 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 opp. the height of the tree to the nearest foot? Which function relates the opposite and the adjacent? 180(tan 38.7°) = x 144 ≈ x h ≈ 150 Substitute 38.7 for θ, x for opp., and 180 for adj. Multiply both sides by 180. Use a calculator to simplify. Add 6 for the biologist’s height. The height of the tree is about 150 ft. Holt McDougal Algebra 2 adj. Right-Angle Trigonometry Geology Application 6. A surveyor whose eye level is 6 ft above the ground measures the angle of elevation to the top of the x highest hill on a roller coaster to be 60.7°. If the opp. surveyor is standing 120 ft from the hill’s base, what is the height of the hill to the nearest foot? Which function relates the opposite and the adjacent? 120(tan 60.7°) = x 60.7° 120 ft adj. Substitute 60.7 for θ, x for opp., and 120 for adj. Multiply both sides by 120. 214 ≈ x Use a calculator to simplify. h ≈ 220 Add 6 for the surveyor’s height. The height of the hill is about 220 ft. Holt McDougal Algebra 2 Right-Angle Trigonometry Lesson 10.1 Practice B Holt McDougal Algebra 2