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8-58-5 LawLaw of of Sines and and Law Law of of Cosines Cosines

Warm Up Lesson Presentation Lesson Quiz

HoltHolt Geometry 8-5 and Law of Cosines

Warm Up 1. What is the third measure in a with measuring 65° and 43°? 72° Find each value. Round trigonometric ratios to the nearest hundredth and angle measures to the nearest . 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) 7. tan-1 (2.75) 20° 51° 70°

Holt Geometry 8-5 Law of Sines and Law of Cosines

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

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° C. sin 93°

tan 103° ≈ –4.33 cos 165° ≈ –0.97 sin 93° ≈ 1.00

Holt Geometry 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° b. cos 92° c. sin 160°

tan 175° ≈ –0.09 cos 92° ≈ –0.03 sin 160° ≈ 0.34

Holt Geometry 8-5 Law of Sines and Law of Cosines

You can use the 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

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° 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. m∠Q Law of Sines

Substitute the given values.

Multiply both sides by 6.

Use the inverse function to find m∠Q.

Holt 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

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 2b

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 2c 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 2d

Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. AC m∠A + m∠B + m∠C = 180° Prop of ∆. m∠A + 67° + 44° = 180° Substitute the given values. m∠A = 69° Simplify.

Holt Geometry 8-5 Law of Sines and Law of Cosines Check It Out! Example 2D 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

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

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

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 Simplify. Find the square XZ ≈ 53.3 root of both sides.

Holt 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∠T

RS2 = RT2 + ST2 – 2(RT)(ST)cos T Law of Cosines Substitute the 2 2 2 7 = 13 + 11 – 2(13)(11)cos T given values. 49 = 290 – 286 cosT Simplify. Subtract 290 –241 = –286 cosT both sides.

Holt 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∠T –241 = –286 cosT Solve for cosT.

Use the inverse cosine function to find m∠T.

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 Law of Cosines Substitute the 2 2 = 18 + 16 – 2(18)(16)cos 21° given values. DE2 ≈ 42.2577 Simplify. Find the square DE ≈ 6.5 root of both sides.

Holt 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∠K JL2 = LK2 + KJ2 – 2(LK)(KJ)cos K Law of Cosines Substitute the 2 2 2 8 = 15 + 10 – 2(15)(10)cos K given values. 64 = 325 – 300 cosK Simplify. Subtract 325 –261 = –300 cosK both sides.

Holt Geometry 8-5 Law of Sines and Law of Cosines 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.

Holt 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 YZ2 = XY2 + XZ2 – 2(XY)(XZ)cos X Law of Cosines Substitute the 2 2 = 10 + 4 – 2(10)(4)cos 34° given values. YZ2 ≈ 49.6770 Simplify. Find the square YZ ≈ 7.0 root of both sides.

Holt 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∠R 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 Simplify. Subtract 145.06 –52.9 = –123.9 cosR both sides.

Holt Geometry 8-5 Law of Sines and Law of Cosines Check It Out! Example 3d Continued Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. m∠R

–52.9 = –123.9 cosR

Solve for cosR.

Use the inverse cosine function to find m∠R.

Holt Geometry 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 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 Geometry 8-5 Law of Sines and Law of Cosines Example 4 Continued

Step 1 Find BC.

BC2 = AB2 + AC2 – 2(AB)(AC)cos A Law of Cosines Substitute the 2 2 = 3.9 + 3.1 – 2(3.9)(3.1)cos 45° given values.

BC2 ≈ 7.7222 Simplify. Find the square BC ≈ 2.8 mi root of both sides.

Holt Geometry 8-5 Law of Sines and Law of Cosines Example 4 Continued

Step 2 Find the measure of the angle through which competitors must turn. This is m∠C.

Law of Sines Substitute the given values.

Multiply both sides by 3.9. Use the inverse sine function to find m∠C.

Holt Geometry 8-5 Law of Sines and Law of Cosines 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

Holt Geometry 8-5 Law of Sines and Law of Cosines 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 2 2 = 31 + 56 – 2(31)(56)cos 100° given values.

AC2 ≈ 4699.9065 Simplify. Find the square AC ≈68.6 m root of both sides.

Holt Geometry 8-5 Law of Sines and Law of Cosines 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.

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. m∠B = 20°, m∠C = 31° and b = 210. Find a. 477.2

5. a = 16, b = 10, and m∠C = 110°. Find c. 21.6

6. a = 20, b = 15, and c = 8.3. Find m∠A. 115°

Holt Geometry 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