Transcript 8-5
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 8-5 Law of Sines and Law of Cosines Holt Geometry 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° tan 103° – 4.33 Holt Geometry B. cos 165° cos 165° – 0.97 C. sin 93° sin 93° 1.00 8-5 Law of Sines and Law of Cosines 8.5 Practice (On a Separate Sheet of Paper) Holt Geometry 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° Holt Geometry Cross Products Property Divide both sides by sin 39. 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 8.5 Practice (Continued) 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 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 Law of Cosines Substitute the 2 2 = 35 + 30 – 2(35)(30)cos 110° given values. XZ2 2843.2423 XZ 53.3 Holt Geometry 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 Check It Out! Example 3d Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. mR PQ2 = PR2 + RQ2 – 2(PR)(RQ)cos R Law of Cosines Substitute the 2 2 2 9.6 = 5.9 + 10.5 – 2(5.9)(10.5)cos R given values. 92.16 = 145.06 – 123.9cosR –52.9 = –123.9 cosR Holt Geometry Simplify. Subtract 145.06 both sides. 8-5 Law of Sines and Law of Cosines 8.5 Practice (Continued) 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° Holt Geometry 0.31 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 477.2 a. 21.6 5. a = 16, b = 10, and mC = 110°. Find c. 6. a = 20, b = 15, and c = 8.3. Find mA. Holt Geometry 115° 8-5 Law of Sines and Law of Cosines Lesson Quiz: Part III 7. An observer in tower A sees a fire 1554 ft away at an angle of depression of 28°. To the nearest foot, how far is the fire from an observer in tower B? To the nearest degree, what is the angle of depression to the fire from tower B? 1212 ft; 37° Holt Geometry