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Warm-Up Exercises 2. Name the leg opposite X. 1. Name the hypotenuse. Use this diagram for Exercises 1-4. ANSWER YZ ANSWER XZ

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Warm-Up Exercises ANSWER 14.78 3. Name the leg adjacent to X. 4. If XY = 17 and m X = 41, find AC. ANSWER XY Use this diagram for Exercises 1-4.

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Warm-Up Exercises EXAMPLE 1 Find sine ratios Find sin S and sin R. Write each answer as a fraction and as a decimal rounded to four places. SOLUTION sin S = opp. S hyp = RT SR = 63 65 0.9692 sin R = opp. R hyp = ST SR = 16 65 0.2462

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

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

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Warm-Up Exercises EXAMPLE 3 Use a trigonometric ratio to find a hypotenuse 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.

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

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

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

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

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

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Warm-Up Exercises EXAMPLE 5 Find leg lengths using an angle of elevation 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.

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

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

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Warm-Up Exercises 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 60 o angle. SOLUTION sin 60 o = opp. hyp. = 3 2 0.08660 cos 60 o = adj. hyp. = 2 1 0.5000 = Use the 30 o - 60 o - 90 o Triangle Theorem to draw a right triangle with side lengths of 1,, and 2. Then set up sine and cosine ratios for the 60 o angle. 3

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Warm-Up Exercises GUIDED PRACTICE for Examples 5 and 6 7.7. WHAT IF? In Example 5, suppose the angle of elevation is 35 o. What is the new height and base length of the ramp? ANSWERabout 8 feet, about 11.5 ft 8. Use a special right triangle to find the sine and cosine of a 30 o angle. ANSWER 1 2, 2 3

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Warm-Up Exercises Daily Homework Quiz Use this diagram for Exercises 1- 3. 1. If x =, y = 4 and z =, find sin X, sin Y, cos X and cos Y. 5 46 4 ANSWER sin X = 30 6 0.9129, sin Y = 0.4082, 6 6 cos Y = 30 6 0.9129 cos X = 0.4082, 6 6

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Warm-Up Exercises Daily Homework Quiz Use this diagram for Exercises 1- 3. ANSWER 38.6 2. If y = 10 and m Y = 15, find z to the nearest tenth. o

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Warm-Up Exercises Daily Homework Quiz Use this diagram for Exercises 1- 3. ANSWER 1.3 3. If z = 12 and m X = 84, find y to the nearest tenth. o

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Warm-Up Exercises Daily Homework Quiz ANSWER 14.6 ft 4. A lamppost is 11 feet tall. If the angle of elevation of the sun is 49 o, how far is the top of the lamppost from the tip of its shadow?

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EXAMPLE 5 Find leg lengths using an angle of elevation SKATEBOARD RAMP You want to build a skateboard ramp with a length of 14 feet and an angle of elevation.

EXAMPLE 5 Find leg lengths using an angle of elevation SKATEBOARD RAMP You want to build a skateboard ramp with a length of 14 feet and an angle of elevation.

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