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Understanding Trigonometric Ratios and the Law of Sines and Cosines

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kass
18 diapositives · 386 Views · Sep 13, 2014

This instructional guide provides step-by-step examples for calculating trigonometric ratios, as well as applying the Law of Sines and the Law of Cosines to find missing side lengths and angles in triangles. Each example includes detailed calculations and rounding instructions for both lengths and angle measures. Learn how to effectively use a calculator for determining values, apply cross-products, and utilize inverse functions to solve for unknowns. Ideal for students looking to enhance their understanding of trigonometry.

#understanding trigonometric ratios and the law of sines and cosines #understanding trigonometric ratios and the law of sines and cosines ppt #powerpoint #presentation
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Understanding Trigonometric Ratios and the Law of Sines and Cosines
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  1. 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

  2. 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°.

  3. 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. Cross Products Property 10 sin L = 6 sin 125° Use the inverse sine function to find mL.

  4. 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. Cross Products Property 7.6 sin X = 4.3 sin 50° Use the inverse sine function to find mX.

  5. 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 ∆. Substitute the given values. mA + 67° + 44° = 180° mA = 69° Simplify.

  6. 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°.

  7. 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 given values. = 182 + 162 – 2(18)(16)cos 21° DE2 42.2577 Simplify. Find the square root of both sides. DE 6.5

  8. 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 given values. 82 = 152 + 102 – 2(15)(10)cos K 64 = 325 – 300 cosK Simplify. Subtract 325 both sides. –261 = –300 cosK

  9. 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.

  10. 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 given values. = 102 + 42 – 2(10)(4)cos 34° YZ2 49.6770 Simplify. Find the square root of both sides. YZ 7.0

  11. 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 given values. 9.62 = 5.92 + 10.52 – 2(5.9)(10.5)cos R 92.16 = 145.06 – 123.9cosR Simplify. Subtract 145.06 both sides. –52.9 = –123.9 cosR

  12. 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.

  13. 31 m 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.

  14. 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

  15. 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.

  16. 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° –0.49 –0.56 0.31

  17. 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. 477.2 21.6 115°

  18. 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°