Holt McDougal Geometry 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°? Find each value. Round trigonometric ratios to the nearest hundredth and angle measures to the nearest degree. 2. sin 73°3. cos 18°4. tan 82° 5. sin -1 (0.34)6. cos -1 (0.63)7. tan -1 (2.75) 72° °51°70°

Holt McDougal Geometry 8-5 Law of Sines and Law of Cosines Use the Law of Sines and the Law of Cosines to solve triangles. Objective

Holt McDougal 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 McDougal 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 McDougal 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

Holt McDougal 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. mQmQ

Holt McDougal Geometry 8-5 Law of Sines and Law of Cosines Check It Out! Example 2a Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. NP

Holt McDougal 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. mLmL

Holt McDougal Geometry 8-5 Law of Sines and Law of Cosines Check It Out! Example 2c Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. mXmX

Holt McDougal Geometry 8-5 Law of Sines and Law of Cosines Check It Out! Example 2d Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. AC

Holt McDougal 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 McDougal 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 McDougal Geometry 8-5 Law of Sines and Law of Cosines The angle referenced in the Law of Cosines is across the equal sign from its corresponding side. Helpful Hint

Holt McDougal 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

Holt McDougal Geometry 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. mTmT

Holt McDougal Geometry 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. mTmT

Holt McDougal 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

Holt McDougal Geometry 8-5 Law of Sines and Law of Cosines Check It Out! Example 3b Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. mKmK

Holt McDougal Geometry 8-5 Law of Sines and Law of Cosines Check It Out! Example 3c Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. YZ

Holt McDougal Geometry 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. mRmR

Holt McDougal Geometry 8-5 Law of Sines and Law of Cosines Example 4: Sailing Application A sailing club has planned a triangular racecourse, as shown in the diagram. How long is the leg of the race along BC? How many degrees must competitors turn at point C? Round the length to the nearest tenth and the angle measure to the nearest degree.

Holt McDougal Geometry 8-5 Law of Sines and Law of Cosines Example 4 Continued Step 1 Find BC. BC 2 = AB 2 + AC 2 – 2(AB)(AC)cos A = – 2(3.9)(3.1)cos 45° BC 2  BC  2.8 mi Law of Cosines Substitute the given values. Simplify. Find the square root of both sides.

Holt McDougal Geometry 8-5 Law of Sines and Law of Cosines Step 2 Find the measure of the angle through which competitors must turn. This is mC. Example 4 Continued Law of Sines Substitute the given values. Multiply both sides by 3.9. Use the inverse sine function to find mC.

Holt McDougal Geometry 8-5 Law of Sines and Law of Cosines Assignment Chapter 8.5 Pg. 573 (1-41 odd)