8 5 Law of Sines and Law of

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 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° 1. 00

8 -5 Law of Sines and Law of Cosines Check It Out! Example 1 Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. a. tan 175° – 0. 09 Holt Geometry b. cos 92° – 0. 03 c. sin 160° 0. 34

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 2 A: 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 2 B: Using the Law of Sines Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. m Q Law of Sines Substitute the given values. Multiply both sides by 6. Use the inverse sine function to find m Q. Holt Geometry

8 -5 Law of Sines and Law of Cosines Check It Out! Example 2 a Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. NP Law of Sines Substitute the given values. NP sin 39° = 22 sin 88° Cross Products Property Divide both sides by sin 39°. Holt Geometry

8 -5 Law of Sines and Law of Cosines Check It Out! Example 2 b Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. m L Law of Sines Substitute the given values. 10 sin L = 6 sin 125° Cross Products Property Use the inverse sine function to find m L. Holt Geometry

8 -5 Law of Sines and Law of Cosines Check It Out! Example 2 c Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. m X Law of Sines Substitute the given values. 7. 6 sin X = 4. 3 sin 50° Cross Products Property Use the inverse sine function to find m X. Holt Geometry

8 -5 Law of Sines and Law of Cosines Check It Out! Example 2 d Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. AC m A + m B + m C = 180° m A + 67° + 44° = 180° m A = 69° Holt Geometry Prop of ∆. Substitute the given values. Simplify.

8 -5 Law of Sines and Law of Cosines Check It Out! Example 2 D Continued Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. Law of Sines Substitute the given values. AC sin 69° = 18 sin 67° Cross Products Property Divide both sides by sin 69°. 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 3 A: Using the Law of Cosines Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. XZ XZ 2 = XY 2 + YZ 2 – 2(XY)(YZ)cos Y = 352 + 302 – 2(35)(30)cos 110° XZ 2 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 Check It Out! Example 3 a Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. DE DE 2 = EF 2 + DF 2 – 2(EF)(DF)cos F = 182 + 162 – 2(18)(16)cos 21° DE 2 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 Example 3 B: Using the Law of Cosines Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. m T RS 2 = RT 2 + ST 2 – 2(RT)(ST)cos T 72 = 132 + 112 – 2(13)(11)cos T 49 = 290 – 286 cos. T – 241 = – 286 cos. T Holt Geometry Law of Cosines Substitute the given values. Simplify. Subtract 290 both sides.

8 -5 Law of Sines and Law of Cosines Example 3 B Continued Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. m T – 241 = – 286 cos. T Solve for cos. T. Use the inverse cosine function to find m T. Holt Geometry

8 -5 Law of Sines and Law of Cosines Check It Out! Example 3 b Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. m K JL 2 = LK 2 + KJ 2 – 2(LK)(KJ)cos K 82 = 152 + 102 – 2(15)(10)cos K 64 = 325 – 300 cos. K – 261 = – 300 cos. K Holt Geometry Law of Cosines Substitute the given values. Simplify. Subtract 325 both sides.

8 -5 Law of Sines and Law of Cosines Check It Out! Example 3 b Continued Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. m K – 261 = – 300 cos. K Solve for cos. K. Use the inverse cosine function to find m K. Holt Geometry

8 -5 Law of Sines and Law of Cosines Check It Out! Example 3 c Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. YZ YZ 2 = XY 2 + XZ 2 – 2(XY)(XZ)cos X = 102 + 42 – 2(10)(4)cos 34° YZ 2 49. 6770 YZ 7. 0 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 Check It Out! Example 3 d Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. m R PQ 2 = PR 2 + RQ 2 – 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. 9 cos. R – 52. 9 = – 123. 9 cos. R Holt Geometry Simplify. Subtract 145. 06 both sides.

8 -5 Law of Sines and Law of Cosines Check It Out! Example 3 d Continued Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. m R – 52. 9 = – 123. 9 cos. R Solve for cos. R. Use the inverse cosine function to find m R. Holt Geometry