EXAMPLE 2 ALGEBRA Find a leg length Find

EXAMPLE 2 ALGEBRA Find a leg length Find the value of x. SOLUTION Use the tangent of an acute angle to find a leg length. opp. tan 32 o = Write ratio for tangent of 32 o. adj. Substitute. tan 32 o = 11 x Multiply each side by x. x tan 32 o = 11 11 x = Divide each side by tan 32 o o tan 32 Use a calculator to find tan 32 o 11 x 0. 6249 x 17. 6 Simplify

EXAMPLE 3 Estimate height using tangent LAMPPOST Find the height h of the lamppost to the nearest inch. opp. tan 70 o = adj. h. 40. tan 70 o = o 40 tan 70 109. 9 Write ratio for tangent of 70 o. Substitute. = h h Multiply each side by 40. Use a calculator to simplify. ANSWER The lamppost is about 110 inches tall.

EXAMPLE 4 Use a special right triangle to find a tangent Use a special right triangle to find the tangent of a 60 o angle. STEP 1 Because all 30 o-60 o-90 o triangles are similar, you can simplify your calculations by choosing 1 as the length of the shorter leg. Use the 30 o-60 o-90 o Triangle Theorem to find the length of the longer leg.

EXAMPLE 4 Use a special right triangle to find a tangent longer leg = shorter leg 3 30 o- 60 o- 90 o Triangle Theorem x =1 3 Substitute. x = 3 Simplify.

EXAMPLE 4 Use a special right triangle to find a tangent STEP 2 tan 60 o = opp. adj. Write ratio for tangent of 60 o. = 3 1 Substitute. tan 60 o = 3 Simplify. o tan 60 ANSWER The tangent of any 60 o angle is 3 1. 7321

GUIDED PRACTICE for Examples 2, 3, and 4 Find the value of x. Round to the nearest tenth. opp. tan 61 o = adj. Write ratio for tangent of 60 o. tan 61 o = 22 x Substitute. x tan 61 o = 22 Multiply each side by x. x = 12. 2 Simplify.

GUIDED PRACTICE for Examples 2, 3, and 4 Find the value of x. Round to the nearest tenth. opp. tan 56 o = adj. Write ratio for tangent of 56 o. tan 56 o = x 13 13 tan 56 o = x Multiply each side by 13. 19. 3 = x Simplify. Substitute.

for Examples 2, 3, and 4 GUIDED PRACTICE Find the value of x. Round to the nearest tenth. 5. What If ? In Example 4, suppose the side length of the shorter leg is 5 instead of 1. Show that the tangent of 60° is still equal to 3 opp. tan 60 o = adj. Write ratio for tangent of 60 o. tan 60 o = 5 3 5 Substitute. tan 60 o = Simplify. ANSWER 3 The tangent of any 60 o angle is 3