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Apply Sine and Cosine Ratios

Apply Sine and Cosine Ratios. 5.3 (M2). B. C. A. Vocabulary. Sine and Cosine ratios: trig. Ratios for acute angles with legs and hypotenuse. SohCahToa. Sine. Cosine. opposite. adjacent. opposite. adjacent. Tangent. hypotenuse. hypotenuse. SohCahToa. old. Some. oats. away.

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Apply Sine and Cosine Ratios

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  1. Apply Sine and Cosine Ratios 5.3 (M2)

  2. B C A Vocabulary • Sine and Cosine ratios: trig. Ratios for acute angles with legs and hypotenuse

  3. SohCahToa Sine Cosine opposite adjacent opposite adjacent Tangent hypotenuse hypotenuse

  4. SohCahToa old Some oats away horse horse taking caught another

  5. opp. S 63 16 RT ST = = = 0.9692 hyp SR 65 SR 65 opp. R = = = 0.2462 hyp EXAMPLE 1 Find sine ratios Find sinSand sinR. Write each answer as a fraction and as a decimal rounded to four places. SOLUTION sinS sinR

  6. ANSWER ANSWER 3 or 0.6, 8 or 0.4706, 5 17 4 or 0.8 15 or 0.8824 5 17 for Example 1 GUIDED PRACTICE Find sin Xand sin Y. Write each answer as a fraction and as a decimal. Round to four decimal places, if necessary.

  7. adj. to W adj. to U = = hyp hyp UV 18 WV 24 = = = = 3 30 UW UW 30 = 5 4 = 5 EXAMPLE 2 Find cosine ratios Find cosU and cosW. Write each answer as a fraction and as a decimal. SOLUTION cosU = 0.6000 cosW = 0.8000

  8. DOG RUN You want to string cable to make a dog run from two corners of a building, as shown in the diagram. Write and solve a proportion using a trigonometric ratio to approximate the length of cable you will need. EXAMPLE 3 Use a trigonometric ratio to find a hypotenuse

  9. sin 35o = opp. hyp. 11 sin 35o = x x sin 35o = 11 x = x 11 11 sin35o 0.5736 ANSWER You will need a little more than 19 feet of cable. x 19.2 EXAMPLE 3 Use a trigonometric ratio to find a hypotenuse SOLUTION Write ratio for sine of 35o. Substitute. Multiply each side by x. Divide each side by sin 35o. Use a calculator to find sin 35o. Simplify.

  10. 0.6, 0.8 0.8824, 0.4706 about 15.7 ft ANSWER ANSWER ANSWER 5. In Example 3, use the cosine ratio to find the length of the other leg of the triangle formed. for Examples 2 and 3 GUIDED PRACTICE In Exercises 3 and 4, find cosRand cosS. Write each answer as a decimal. Round to four decimal places, if necessary.

  11. You are skiing on a mountain with an altitude of 1200 meters. The angle of depression is 21o. About how far do you ski down the mountain? EXAMPLE 4 Find a hypotenuse using an angle of depression SKIING

  12. = opp. hyp. 1200 = x x sin 21o 1200 x = sin21o 1200 x 0.3584 ANSWER You ski about 3348 meters down the mountain. x 3348.2 EXAMPLE 4 Find a hypotenuse using an angle of depression SOLUTION sin 21o Write ratio for sine of 21o. sin 21o Substitute. = 1200 Multiply each side by x. Divide each side by sin 21o Use a calculator to find sin21o Simplify.

  13. 6. WHAT IF? Suppose the angle of depression in Example 4 is 28°. About how far would you ski? about 2556 m ANSWER for Example 4 GUIDED PRACTICE

  14. SKATEBOARD RAMP You want to build a skateboard ramp with a length of 14 feet and an angle of elevation of 26°. You need to find the height and length of the base of the ramp. EXAMPLE 5 Find leg lengths using an angle of elevation

  15. Find the height. STEP 1 sin26o = x = opp. hyp. sin26o 14 14 sin 26o = x 6.1 x ANSWER The height is about 6.1 feet. EXAMPLE 5 Find leg lengths using an angle of elevation SOLUTION Write ratio for sine of 26o. Substitute. Multiply each side by 14. Use a calculator to simplify.

  16. cos26o = cos26o y = adj. hyp. 14 14 cos 26o = y 12.6 y ANSWER The length of the base is about 12.6 feet. EXAMPLE 5 Find leg lengths using an angle of elevation Find the length of the base. STEP 2 Write ratio for cosine of 26o. Substitute. Multiply each side by 14. Use a calculator to simplify.

  17. 3 3 Use the 30o - 60o - 90o Triangle Theorem to draw a right triangle with side lengths of 1, , and 2. Then set up sine and cosine ratios for the 60oangle. adj. hyp. opp. hyp. sin60o = 0.08660 = 2 1 cos60o = = 0.5000 = 2 EXAMPLE 6 Use a special right triangle to find a sine and cosine Use a special right triangle to find the sine and cosine of a 60o angle. SOLUTION

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