#### Transcript 8-5 - Nutley Public Schools

8-5 8-5 Law LawofofSines Sinesand andLaw LawofofCosines Cosines Warm Up Lesson Presentation Lesson Quiz Holt Geometry Holt 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°? 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 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 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° 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 Check It Out! Example 1 Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. a. tan 175° b. cos 92° tan 175° –0.09 cos 92° –0.03 Holt Geometry c. sin 160° 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 • AAS • two side lengths and a non-included angle measure • SSA :O Holt Geometry 8-5 Law of Sines and Law of Cosines The Law of Sines is a pain in the SSA, when it comes to SSA triangles. • When you have an SSA triangle, there CAN be • No solutions • 1 solution • 2 solutions 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 This is an SSA triangle! Lets see how many solutions there are. Do the problem as you normally would.. 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. 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 There is definitely 1 solution. Lets check if there is another solution. Subtract the answer you got for <Q from 180. 180-36= 144 Can 151 be a solution for <Q? No! So there is 1 solution and it is 36 degrees 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. mL This is an SSA triangle! Lets see how many solutions there are. Do the problem as you normally would.. 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. mL Law of Sines Substitute the given values. 10 sin L = 6 sin 125° Cross Products Property Use the inverse sine function to find mL. Holt Geometry 8-5 Law of Sines and Law of Cosines There is definitely 1 solution. Lets check if there is another solution. Subtract the answer you got for <Q from 180. 180-29= 151 Can 151 be a solution for <Q? No! So there is 1 solution and it is 29 degrees 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 mA + mB + mC = 180° Prop of ∆. mA + 67° + 44° = 180° Substitute the given values. mA = 69° Holt Geometry Simplify. 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° Holt Geometry Cross Products Property Divide both sides by sin 69°. 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 • 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 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 Example 3B Continued Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. mT –241 = –286 cosT Solve for cosT. Use the inverse cosine function to find mT. 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 = 182 + 162 – 2(18)(16)cos 21° DE2 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 Check It Out! Example 3b Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. mK JL2 = LK2 + KJ2 – 2(LK)(KJ)cos K 82 = 152 + 102 – 2(15)(10)cos K 64 = 325 – 300 cosK –261 = –300 cosK 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 3b Continued Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. mK –261 = –300 cosK Solve for cosK. Use the inverse cosine function to find mK. 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 = 102 + 42 – 2(10)(4)cos 34° YZ2 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 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 Check It Out! Example 3d Continued Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. mR –52.9 = –123.9 cosR Solve for cosR. Use the inverse cosine function to find mR. 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 3.92 = + 45° BC2 7.7222 BC 2.8 mi Holt Geometry 3.12 – 2(3.9)(3.1)cos 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 4 Continued Step 2 Find the measure of the angle through which competitors must turn. This is mC. Law of Sines Substitute the given values. Multiply both sides by 3.9. Use the inverse sine function to find mC. 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 = 312 + 562 Substitute the – 2(31)(56)cos 100° given values. AC2 4699.9065 AC 68.6 m Holt Geometry Law of Cosines Simplify. Find the square root of both sides. 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 mA. 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