Find the values of the variables. 1. ANSWER x = 2 6, y = 2 15

Find the values of the variables. 2. ANSWER 15 2

EXAMPLE 1 Find tangent ratios Find tan S and tan R. Write each answer as a fraction and as a decimal rounded to four places. SOLUTION = opp S adj. to S = RTST = 80 18 = 40 9 tan S 4.4444 = opp R adj. to R = STRT = 18 80 = 9 40 tan R 0.2250 =

GUIDED PRACTICE for Example 1 Find tan J and tan K. Round to four decimal places. ANSWER 0.7500, 1.3333 ANSWER 0.5333, 1.8750

Use the tangent of an acute angle to find a leg length. EXAMPLE 2 Find a leg length ALGEBRA Find the value of x. SOLUTION Use the tangent of an acute angle to find a leg length. tan 32o = opp. adj. Write ratio for tangent of 32o. tan 32o 11 = x Substitute. x tan 32o = 11 Multiply each side by x. x = 11 tan 32o Divide each side by tan 32o x 11 0.6249 Use a calculator to find tan 32o x 17.6 Simplify

Estimate height using tangent EXAMPLE 3 Estimate height using tangent LAMPPOST Find the height h of the lamppost to the nearest inch. tan 70o = opp. adj. Write ratio for tangent of 70o. tan 70o h = 40 Substitute. Multiply each side by 40. 40 tan 70o = h Use a calculator to simplify. 109.9 h 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 60o angle. STEP 1 Because all 30o-60o-90o triangles are similar, you can simplify your calculations by choosing 1 as the length of the shorter leg. Use the 30o-60o-90o Triangle Theorem to find the length of the longer leg.

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

Use a special right triangle to find a tangent EXAMPLE 4 Use a special right triangle to find a tangent STEP 2 Find tan 60o tan 60o = opp. adj. Write ratio for tangent of 60o. tan 60o = 3 1 Substitute. tan 60o = 3 Simplify. ANSWER The tangent of any 60o angle is 3 1.7321

Find the value of x. Round to the nearest tenth. GUIDED PRACTICE for Examples 2, 3, and 4 Find the value of x. Round to the nearest tenth. ANSWER 12.2 ANSWER 19.3 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 . 5. 3 ANSWER shorter leg = 5, longer leg = , 3 5 tan 60 =

Daily Homework Quiz Use this diagram for Exercises 1- 4. 1. If a = 18, b = 80 and c = 82, find tan B and tan A. Write each answer as a fraction and a decimal rounded to 4 places. ANSWER tan B = 40 9 = 4.4444; tan A = = 0.2250

Use this diagram for Exercises 1- 4. Daily Homework Quiz Use this diagram for Exercises 1- 4. 2. If a = 17 and m A = 31 , find b to the nearest tenth. o ANSWER 28.3

Use this diagram for Exercises 1- 4. Daily Homework Quiz Use this diagram for Exercises 1- 4. 3. If a = 9 and m B = 74 , find b to the nearest tenth. o ANSWER 31.4

Use this diagram for Exercises 1- 4. Daily Homework Quiz Use this diagram for Exercises 1- 4. 4. If a = 5 and m B = 79 , find the number of square units in the area of ABC to the nearest tenth. o ANSWER 64.3