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Name ________________________________________ Date __________________ Class __________________ LESSON 13-1 Tangent Ratio Practice and Problem Solving: A/B Identify the relationships in the figure to the right. 1. tanX = 3. tan−1 WX = VW = m∠_____ WX 5. tanX × tanV = _____ 2. tanV = 4. tan−1 WX = m∠_____ VW 6. tan−1 VW + tan−1 WX = _____° WX VW Use a calculator to find each tangent or inverse tangent. Round tangents to the nearest 0.01 and angles to the nearest 0.1 degree. Check the inverse tangents by finding the tangent of each angle. 7. tan23° ≈ _____________ 10. tan−10.14 ≈ _____________° tan _____________° ≈ 0.14 8. tan43° ≈ _____________ 11. tan−11= _____________° 9. tan47° ≈ _____________ 12. tan−16.1 ≈ _____________° tan _____________° = 1 tan _____________° ≈ 6.1 Solve Problems 13–16 using tangent ratios and a calculator. Refer to the figure to the right of each problem. 13. To the nearest hundredth, what is tanM in +LMN ? ________ 14. Write a ratio that gives tanS. ________ Find the value of tanS to the nearest hundredth. ________ Use the inverse tangent function on your calculator to find the angle with that tangent. ________ 15. Write and solve a tangent equation to find the distance from C to E to the nearest 0.1 meter. ________ meters 16. The glide slope is the path a plane uses while it is landing on a runway. The glide slope usually makes a 3° angle with the ground. A plane is on the glide slope and is 1 mile (5280 feet) from touchdown. Find EF, the plane’s altitude, to the nearest foot. Show your work. _________________________________________________________________________________________ _________________________________________________________________________________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 253 Name ________________________________________ Date __________________ Class __________________ LESSON 13-2 Sine and Cosine Ratios Practice and Problem Solving: A/B After verifying that the triangle to the right is a right triangle, use a calculator to find the given measures. Give ratios to the nearest hundredth and angles to the nearest degree. 1. Use the Pythagorean Theorem to confirm that the triangle is a right triangle. Show your work. ________________________________________________________________ 2. sin∠1 = ≈ _________________ 3. sin∠2 = 4. cos∠1 = = _________________ 5. cos∠2 = = _______________________ ≈ _______________________ 6. Show how to find m∠1 using the inverse sine of ∠1. _________________________________________________________________________________________ 7. Show how to find m∠2 using the inverse sine of ∠2. _________________________________________________________________________________________ Use a calculator and trigonometric ratios to find each length. Round to the nearest hundredth. 8. 9. BD = _________________ 10. QP = _________________ ST = _________________ Use sine and cosine ratios to solve Problems 11–13. 11. Find the perimeter of the triangle. Round to the nearest 0.1 centimeter. _________________ 12. To the nearest 0.1 inch, what is the length of the hypotenuse of the springboard shown to the right? _________________ 13. What is the height of the springboard (the dotted line)? _________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 258 Name ________________________________________ Date __________________ Class __________________ LESSON 13-3 Special Right Triangles Practice and Problem Solving: A/B Use the figure to the right for Problems 1−4. Write each trigonometric ratio as a simplified fraction and as a decimal rounded to the nearest hundredth. 1. sinL _________________ 3. tanM _________________ 2. cosL _________________ 4. sinM _________________ Write each trigonometric ratio as a simplified fraction. 5. sin 30° = _______________ 6. cos 30° = _______________ 7. tan 45° = _______________ 8. tan 30° = _________________ 9. cos 45° = _______________ 10. tan 60° = _______________ 11. Fill in the side lengths for these special right triangles with a hypotenuse of 1. Use decimals to the nearest 0.01, and be sure that your answers make sense, for example that the hypotenuse is longer than the legs. Use special right triangle relationships to solve Problems 12–14. 12. If cos A = 0.28, which angle in the triangles to the right is ∠A? _______________ If sin B = 0.22, which angle is ∠B? _______________ 13. What is EF, the measure of the longest side of the sail on the model? Round to the nearest inch. _________________ in. What is the measure of the shortest side? _________________ in. 14. If the small sail is similar to the larger one and is 11 inches high, about how wide is it? _________________ in. Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 263 Name ________________________________________ Date __________________ Class __________________ LESSON 13-4 Problem Solving with Trigonometry Practice and Problem Solving: C For Problems 1–6, use trigonometry and the Pythagorean theorem to solve the right triangles on the coordinate plane. Show your work. 1. First use the slope formula to verify that +ABC is a right triangle. ____________________________________ 2. Use the distance formula to find the length of each side. AB = ________ BC = ________ AC = ________ 3. Use the Pythagorean theorem to double check the side lengths. ____________________________ 4. Use inverse trigonometric ratios to find the acute angles. m∠A = ________ m∠C = ________ + 5. Verify that PQR is a right triangle. Find the three side lengths and the measures of the acute angles. PQ = ________ QR = ________ m∠P = ________ m∠Q = ________ RP = ________ 6. Find the side lengths and angle measures for X(1, 0), Y(2, 1), Z(5, −2). +XYZ, XY = ________ YZ = ________ XZ = ________ m∠X = ________ m∠Y = ________ m∠Z = ________ For Problems 7–10, use trigonometric functions to find the area of the triangles, to the nearest square unit. 7. If you know the lengths of two sides of any triangle, a and b, and the measure of the included angle, m∠C, how can you find the area of the triangle? _________________________________ 8. Find the area of +ABC on the coordinate plane above. _________________________________________________________________________________________ 9. Find the area of +PQR on the coordinate plane above. _________________________________________________________________________________________ 10. Find the area of +XYZ in Problem 6 above. _________________ Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor. 269