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Solving Right Triangles Warm Up Use ∆ABC for Exercises 1–3. 1. If a = 8 and b = 5, find c. 2. If a = 60 and c = 61, find b. 11 3. If b = 6 and c = 10, find sin B. 0.6 Find AB. Distance formula: √(x2 – x1)2+(y2 – y1) 4. A(8, 10), B(3, 0) 5. A(1, –2), B(2, 6) Holt McDougal Geometry Solving Right Triangles Essential Question • How can I use SOHCAHTOA to solve for side lengths and angles? Holt McDougal Geometry Solving Right Triangles Unit 2 Right triangles • Section 4: Solving right triangles Lesson 44 Holt McDougal Geometry Solving Right Triangles Standard(s): • MCC9-12.G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems • MCC9-12.G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Holt McDougal Geometry Solving Right Triangles Example 3B: Calculating Trigonometric Ratios Use your calculator to find the trigonometric ratio. Round to the nearest hundredth. cos 19° cos 19° 0.95 Holt McDougal Geometry Solving Right Triangles Example 3C: Calculating Trigonometric Ratios Use your calculator to find the trigonometric ratio. Round to the nearest hundredth. tan 65° tan 65° 2.14 Holt McDougal Geometry Solving Right Triangles Example 4A: Using Trigonometric Ratios to Find Lengths Find the length. Round to the nearest hundredth. BC is adjacent to the given angle, B. You are given AC, which is opposite B. Since the adjacent and opposite legs are involved, use a tangent ratio. Holt McDougal Geometry Solving Right Triangles Example 4A Continued Write a trigonometric ratio. Substitute the given values. Multiply both sides by BC and divide by tan 15°. BC 38.07 ft Holt McDougal Geometry Simplify the expression. Solving Right Triangles Example 4B: Using Trigonometric Ratios to Find Lengths Find the length. Round to the nearest hundredth. QR is opposite to the given angle, P. You are given PR, which is the hypotenuse. Since the opposite side and hypotenuse are involved, use a sine ratio. Holt McDougal Geometry Solving Right Triangles Example 4B Continued Write a trigonometric ratio. Substitute the given values. 12.9(sin 63°) = QR 11.49 cm QR Holt McDougal Geometry Multiply both sides by 12.9. Simplify the expression. Solving Right Triangles If you know the sine, cosine, or tangent of an acute angle measure, you can use the inverse trigonometric functions to find the measure of the angle. Holt McDougal Geometry Solving Right Triangles Example 2: Calculating Angle Measures from Trigonometric Ratios Use your calculator to find each angle measure to the nearest degree. A. cos-1(0.87) B. sin-1(0.85) C. tan-1(0.71) cos-1(0.87) 30° sin-1(0.85) 58° tan-1(0.71) 35° Holt McDougal Geometry Solving Right Triangles Example 3: Solving Right Triangles Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. Method 1: By the Pythagorean Theorem, RT2 = RS2 + ST2 (5.7)2 = 52 + ST2 Since the acute angles of a right triangle are complementary, mT 90° – 29° 61°. Holt McDougal Geometry Solving Right Triangles Example 4: Solving a Right Triangle in the Coordinate Plane The coordinates of the vertices of ∆PQR are P(–3, 3), Q(2, 3), and R(–3, –4). Find the side lengths to the nearest hundredth and the angle measures to the nearest degree. Holt McDougal Geometry Solving Right Triangles Example 4 Continued Step 1 Find the side lengths. Plot points P, Q, and R. PR = 7 Y P By the Distance Formula, Q X R Holt McDougal Geometry PQ = 5 Solving Right Triangles Example 4 Continued Step 2 Find the angle measures. Y P mP = 90° Q X R The acute s of a rt. ∆ are comp. mR 90° – 54° 36° Holt McDougal Geometry Solving Right Triangles Example 5: Travel Application A highway sign warns that a section of road ahead has a 7% grade. To the nearest degree, what angle does the road make with a horizontal line? Change the percent grade to a fraction. A 7% grade means the road rises (or falls) 7 ft for every 100 ft of horizontal distance. Draw a right triangle to represent the road. A is the angle the road makes with a horizontal line. Holt McDougal Geometry Solving Right Triangles Check It Out! Example 5 Baldwin St. in Dunedin, New Zealand, is the steepest street in the world. It has a grade of 38%. To the nearest degree, what angle does Baldwin St. make with a horizontal line? Change the percent grade to a fraction. A 38% grade means the road rises (or falls) 38 ft for every 100 ft of horizontal distance. C 38 ft A 100 ft B Draw a right triangle to represent the road. A is the angle the road makes with a horizontal line. Holt McDougal Geometry