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Find the values of the variables.

Practice problems and solutions for finding values of variables using tangent ratios and special right triangles. Includes step-by-step examples and guided practice exercises.

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Find the values of the variables.

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

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

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

  4. 0.7500, 1.3333 0.5333, 1.8750 ANSWER ANSWER for Example 1 GUIDED PRACTICE Find tan Jand tan K. Round to four decimal places.

  5. x x 17.6 tan32o = opp. adj. tan32o 11 = 11 = x x = x tan 32o 11 11 tan32o 0.6249 EXAMPLE 2 Find a leg length Find the value of x. ALGEBRA SOLUTION Use the tangent of an acute angle to find a leg length. Write ratio for tangent of 32o. Substitute. Multiply each side by x. Divide each side by tan 32o Use a calculator to find tan 32o Simplify

  6. tan70o = h tan70o = opp. adj. 40 40 tan 70o = h 109.9 h ANSWER The lamppost is about 110 inches tall. EXAMPLE 3 Estimate height using tangent LAMPPOST Find the height hof the lamppost to the nearest inch. Write ratio for tangent of 70o. Substitute. Multiply each side by 40. Use a calculator to simplify.

  7. STEP1 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. EXAMPLE 4 Use a special right triangle to find a tangent Use a special right triangle to find the tangent of a 60o angle.

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

  9. 3 3 3 tan60o = tan60o = 1 opp. adj. tan60o = ANSWER The tangent of any 60o angle is 1.7321 EXAMPLE 4 Use a special right triangle to find a tangent STEP2 Find tan 60o Write ratio for tangent of 60o. Substitute. Simplify.

  10. 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. 5 3 3 12.2 19.3 ANSWER ANSWER ANSWER 5 3 tan60 shorter leg = 5, longer leg = , = = 5 for Examples 2, 3, and 4 GUIDED PRACTICE Find the value of x. Round to the nearest tenth.

  11. 9 Use this diagram for Exercises 1- 4. 40 40 9 1. If a = 18, b = 80 and c = 82, find tan Band tan A. Write each answer as a fraction and a decimal rounded to 4places. ANSWER = 4.4444; = 0.2250 tanB = tanA = Daily Homework Quiz

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

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

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

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