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8-5 Law of Sines and Law of Cosines Warm Up 1. What is the third angle measure in a triangle with angles measuring 65° and 43°? 72° Find each value. Round trigonometric ratios to the nearest hundredth and angle measures to the nearest degree. 2. sin 73° 0.96 3. cos 18° 0.95 4. tan 82° 7.12 5. sin-1 (0.34) 6. cos-1 (0.63) 20° Holt Geometry 51° 7. tan-1 (2.75) 70° 8-5 Law of Sines and Law of Cosines Objective Use the Law of Sines and the Law of Cosines to solve triangles. Holt Geometry 8-5 Law of Sines and Law of Cosines In this lesson, you will learn to solve any triangle. To do so, you will need to calculate trigonometric ratios for angle measures up to 180°. You can use a calculator to find these values. Holt Geometry 8-5 Law of Sines and Law of Cosines Example 1: Finding Trigonometric Ratios for Obtuse Angles Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. A. tan 103° B. cos 165° tan 103° –4.33 cos 165° –0.97 Holt Geometry C. sin 93° sin 93° 1.00 8-5 Law of Sines and Law of Cosines You can use the Law of Sines to solve a triangle if you are given • two angle measures and any side length (ASA or AAS) or • two side lengths and a non-included angle measure (SSA). Holt Geometry 8-5 Law of Sines and Law of Cosines Example 2A: Using the Law of Sines Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. FG Law of Sines Substitute the given values. FG sin 39° = 40 sin 32° Cross Products Property Divide both sides by sin 39. Holt Geometry 8-5 Law of Sines and Law of Cosines Example 2B: Using the Law of Sines Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. mQ Law of Sines Substitute the given values. Multiply both sides by 6. Use the inverse sine function to find mQ. Holt Geometry 8-5 Law of Sines and Law of Cosines Check It Out! Example 2b Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. mL Law of Sines Substitute the given values. 10 sin L = 6 sin 125° Cross Products Property Use the inverse sine function to find mL. Holt Geometry 8-5 Law of Sines and Law of Cosines The Law of Sines cannot be used to solve every triangle. If you know two side lengths and the included angle measure or if you know all three side lengths, you cannot use the Law of Sines. Instead, you can apply the Law of Cosines. Holt Geometry 8-5 Law of Sines and Law of Cosines You can use the Law of Cosines to solve a triangle if you are given • two side lengths and the included angle measure (SAS) or • three side lengths (SSS). Holt Geometry 8-5 Law of Sines and Law of Cosines Example 3A: Using the Law of Cosines Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. XZ XZ2 = XY2 + YZ2 – 2(XY)(YZ)cos Y = 352 + 302 – 2(35)(30)cos 110° XZ2 2843.2423 XZ 53.3 Holt Geometry Law of Cosines Substitute the given values. Simplify. Find the square root of both sides. 8-5 Law of Sines and Law of Cosines Example 3B: Using the Law of Cosines Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. mT RS2 = RT2 + ST2 – 2(RT)(ST)cos T 72 = 132 + 112 – 2(13)(11)cos T 49 = 290 – 286 cosT –241 = –286 cosT Holt Geometry Law of Cosines Substitute the given values. Simplify. Subtract 290 both sides. 8-5 Law of Sines and Law of Cosines Example 3B Continued Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. mT –241 = –286 cosT Solve for cosT. Use the inverse cosine function to find mT. Holt Geometry 8-5 Law of Sines and Law of Cosines Check It Out! Example 3a Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. DE DE2 = EF2 + DF2 – 2(EF)(DF)cos F = 182 + 162 – 2(18)(16)cos 21° DE2 42.2577 DE 6.5 Holt Geometry Law of Cosines Substitute the given values. Simplify. Find the square root of both sides. 8-5 Law of Sines and Law of Cosines Helpful Hint Do not round your answer until the final step of the computation. If a problem has multiple steps, store the calculated answers to each part in your calculator. Holt Geometry 8-5 Law of Sines and Law of Cosines Lesson Quiz: Part I Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. 1. tan 154° –0.49 2. cos 124° –0.56 3. sin 162° 0.31 Holt Geometry 8-5 Law of Sines and Law of Cosines Lesson Quiz: Part II Use ΔABC for Items 4–6. Round lengths to the nearest tenth and angle measures to the nearest degree. 4. mB = 20°, mC = 31° and b = 210. Find a. 477.2 5. a = 16, b = 10, and mC = 110°. Find c. 21.6 6. a = 20, b = 15, and c = 8.3. Find mA. 115° Holt Geometry