Use the diagram for Exercises 1-4. 1. Name the hypotenuse. ANSWER XZ 2. Name the leg opposite X. ANSWER YZ 3. Name the leg adjacent to X. ANSWER XY 4. If XY = 17 and m X = 41 , find YZ. ANSWER 14.78

Use trigonometric ratios to solve right triangles. Target Use trigonometric ratios to solve right triangles. You will… Apply sine and cosine ratios.

tan(reference angle) = opposite side adjacent side Vocabulary tangent ratio – tan(reference angle) = opposite side adjacent side sine ratio – sin(reference angle) = opposite side hypotenuse cosine ratio – cos(reference angle) = adjacent side hypotenuse 3

S = sine C = cosine T = tangent Vocabulary S = sine C = cosine T = tangent O = opposite H = hypotenuse A = adjacent old indian chief Soh Cah Toa Oscar Had A Hold On Arthur Oh Heck Another Heap Of Apples Some Old Horse Caught Another Horse Taking Oats Away 4

tan(reference angle) = opposite side adjacent side Vocabulary tangent ratio – tan(reference angle) = opposite side adjacent side sine ratio – sin(reference angle) = opposite side hypotenuse cosine ratio – cos(reference angle) = adjacent side hypotenuse 5

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 = opp. S hyp = opp. R hyp sin S sin R = RT SR = ST SR sin S sin R = 63 65 = 16 65 sin S sin R sin S 0.9692 sin R 0.2462

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

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. ANSWER ANSWER 8 17 sinX = 15 17 sinY = 15 25 sinX = 20 25 sinY = 0.4706 sinX 0.8824 sinY sinX 0.6000 = sinY 0.8000 =

GUIDED PRACTICE for Example 2 Find cos X and cos Y. Write each answer as a fraction and as a decimal. Round to four decimal places, if necessary. ANSWER ANSWER 15 17 cosX = 8 17 cosY = 20 25 cosX = 15 25 cosY = 0.8824 cosX 0.4706 cosY cosX 0.8000 = cosY 0.6000 =

GUIDED PRACTICE Review Find tan X and tan Y. Write each answer as a fraction and as a decimal. Round to four decimal places, if necessary. ANSWER ANSWER 8 15 tanX = 15 8 tanY = 15 20 tanX = 20 15 tanY = 0.5333 tanX tanY 1.8750 = tanX 0.7500 = 1.3333 tanY

Use a trigonometric ratio to find a hypotenuse 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. 35° x ft 11 ft SOLUTION First you will need to determine which trigonometric ratio is appropriate for the given information. 11 ? tan(35)= 11 x sin(35)= ? x cos(35)=

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

Vocabulary angle of depression – the angle formed by a horizontal line and your line of sight when looking down upon an object angle of elevation – the angle formed by a horizontal line and your line of sight when looking up at on object 14

Find a hypotenuse using an angle of depression 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 21o. About how far do you ski down the mountain? angle of depression SOLUTION Determine which trigonometric ratio to use. 1200 ? tan(21)= 1200 x sin(21)= ? x cos(21)=

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

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. SOLUTION First you will need to determine which trigonometric ratio is appropriate for the given information. x y tan(26)= x 14 sin(26)= y 14 cos(26)=

EXAMPLE 5 Find leg lengths using an angle of elevation SOLUTION Find the height. Find the length of the base. sin 26o = opp. hyp. cos 26o = adj. hyp. sin 26o x = 14 cos 26o y = 14 14 sin 26o = x 14 cos 26o = y 6.1 x 12.6 y ANSWERS The height is about 6.1 feet. The length of the base is about 12.6 feet.