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The Law of Sines and Cosines Chapter 8 – Lesson 5 The Law of Sines and Cosines

Objective Use the Law of Sines and the Law of Cosines to solve triangles.

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.

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° C. sin 93° tan 103°  –4.33 cos 165°  –0.97 sin 93°  1.00

You can use the altitude of a triangle to find a relationship between the triangle’s side lengths. In ∆ABC, let h represent the length of the altitude from C to From the diagram, , and By solving for h, you find that h = b sin A and h = a sin B. So b sin A = a sin B, and . You can use another altitude to show that these ratios equal

You can use the Law of Sines to solve a ▲ if you are given • two ∠ measures and any side length (ASA or AAS) or • two side lengths and a non-included ∠ measure (SSA).

Example 2A: Using the Law of Sines Find . Round to the nearest tenth. FG Use the Law of Sines FG sin 39° = 40 sin 32°

Example 2B: Using the Law of Sines Find the mQ. Round to the nearest degree. Use the Law of Sines          

Check It Out! Example 2D Continued Find AC. Round to the nearest tenth. Law of Sines AC sin 69° = 18 sin 67°

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.

You can use the Law of Cosines to solve a triangle if you are given • two side lengths and the included ∠ measure (SAS) or • three side lengths (SSS).

The angle referenced in the Law of Cosines is across the equal sign from its corresponding side. Helpful Hint

Example 3A: Using the Law of Cosines Find XZ. Round to the nearest tenth. Use the Law of Cosines XZ2 = XY2 + YZ2 – 2(XY)(YZ)cos Y = 352 + 302 – 2(35)(30)cos 110° XZ2  2843.2423 XZ  53.3

Example 3B: Using the Law of Cosines Find the mT. Round to the nearest degree. Use the Law of Cosines RS2 = RT2 + ST2 – 2(RT)(ST)cos T 72 = 132 + 112 – 2(13)(11)cos T 49 = 169 + 121 – 286 cos T –241 = –286 cos T

Check It Out! Example 3b Continued Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. mK –261 = –300 cosK Solve for cosK. Use the inverse cosine function to find mK.

Do not round your answer until the final step of the computation 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. Helpful Hint

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. Law of Sines Law of Cosines BC2 = AB2 + AC2 – 2(AB)(AC)cos A = 3.92 + 3.12 – 2(3.9)(3.1)cos 45° BC2  7.7222 BC  2.8 mi

Check It Out! Example 4 What if…? Another engineer suggested using a cable attached from the top of the tower to a point 31 m from the base. How long would this cable be, and what angle would it make with the ground? Round the length to the nearest tenth and the angle measure to the nearest degree. 31 m

Check It Out! Example 4 Continued Step 1 Find the length of the cable. AC2 = AB2 + BC2 – 2(AB)(BC)cos B Law of Cosines Substitute the given values. = 312 + 562 – 2(31)(56)cos 100° Simplify. AC2  4699.9065 Find the square root of both sides. AC 68.6 m

Check It Out! Example 4 Continued Step 2 Find the measure of the angle the cable would make with the ground. Law of Sines Substitute the given values. Multiply both sides by 56. Use the inverse sine function to find mA.

Lesson Summary: Objective Use the Law of Sines and the Law of Cosines to solve triangles.

Preview of the Next Lesson: Objective Find the magnitude and direction of a vector. Use vectors and vector addition to solve real world problems.

Kahoot!

Stand Up Please

Lesson Quiz: Part I Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. 1. tan 154° 2. cos 124° 3. sin 162°

Lesson Quiz: Part II Use ΔABC for Items 4–6. Round lengths to the nearest tenth and angle measures to the nearest degree. 4. mB = 20°, mC = 31° and b = 210. Find a. 5. a = 16, b = 10, and mC = 110°. Find c. 6. a = 20, b = 15, and c = 8.3. Find mA.  

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? BC2 = AB2 + AC2 – 2(AB)(AC)cos A = 23402 + 15542 – 2(2340)(1554)cos 28°