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

The reciprocals of the sine, cosine, and tangent ratios are also trigonometric ratios. They are trigonometric functions, cosecant, secant, and cotangent.

Finding All Trigonometric Functions Find the values of the six trigonometric functions for θ. Find the length of the hypotenuse. 70 24 θ a2 + b2 = c2 Pythagorean Theorem. hyp. c2 = 242 + 702 Substitute 24 for a and 70 for b. 74 c2 = 5476 Simplify. opp. c = 74 Solve for c. Eliminate the negative solution. Find the lengths of the 6 trigonometric values. adj.

In each reciprocal pair of trigonometric functions, there is exactly one “co” Helpful Hint

Finding All Trigonometric Functions Find the values of the six trigonometric functions for θ. Find the length of the hypotenuse. 80 18 θ a2 + b2 = c2 Pythagorean Theorem. hyp. c2 = 182 + 802 Substitute 18 for a and 80 for b. 82 c2 =6724 Simplify. opp. c = 82 Solve for c. Eliminate the negative solution. Find the lengths of the 6 trigonometric values. adj.

The height above the water is about 5 ft. 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? hyp. opp. 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.

The length of the ramp is about 41 in. Sports Application 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. Use a calculator to simplify. l ≈ 41 The length of the ramp is about 41 in.

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.

The height of the tree is about 150 ft. 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? opp. Which function relates the opposite and the adjacent? adj. Substitute 38.7 for θ, x for opp., and 180 for adj. 180(tan 38.7°) = x Multiply both sides by 180. 144 ≈ x Use a calculator to simplify. h ≈ 150 Add 6 for the biologist’s height. The height of the tree is about 150 ft.

The height of the hill is about 220 ft. Geology Application 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? 120 ft 60.7° x opp. adj. Which function relates the opposite and the adjacent? Substitute 60.7 for θ, x for opp., and 120 for adj. 120(tan 60.7°) = x 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.

Lesson 10.1 Practice B