Warm-Up Determine whether the following triangles are acute, right or obtuse. 1. 7, 10, , 8, , 5, 6

Trigonometry Functions Apply the Sine, Cosine and Tangent Ratios

Vocabulary Trigonometric Ratio: A ratio of the lengths of two sides in a right triangle. Sine: A trigonometric ratio, abbreviated as sin. Tangent: A trigonometric ratio, abbreviated as tan. Cosine: A trigonometric ratio, abbreviated as cos.

Vocabulary Angle of elevation: When you look up at an object, the angle that your line of sight makes with a line drawn horizontally. Angle of depression: When you look down at an object, the angle that your line of sight makes with a line drawn horizontally.

Sine Ratio Sine Ratio: Let ABC be a right triangle with acute A. The sine of A (written as sin A) is defined as follows : Leg Opposite A A sin A = length of leg opposite A length of the hypotenuse hypotenuse B C = BC AB

Cosine Ratio Cosine Ratio: Let ABC be a right triangle with acute A. The cosine of A (written as cos A) is defined as follows : Leg adjacent to A A hypotenuse B C cos A = length of leg adjacent to A length of the hypotenuse = AC AB

Tangent Ratio Tangent Ratio: Let ABC be a right triangle with acute A. The tangent of A (written as tan A) is defined as follows : Leg adjacent to A Leg Opposite A A tan A = length of leg opposite A length of leg adjacent to A hypotenuse B C = BC AC

S O H C A H T O A SineSine OppositeOpposite OppositeOpposite HypotenuseHypotenuse HypotenuseHypotenuse CoSineCoSine TangentTangent AdjacentAdjacent AdjacentAdjacent

Example 1 Find the value of x. Multiply both sides by x. 17° 9 x x tan 17° = 9 Divide both sides by tan 17° tan 17° = opp. adj. tan 17° = 9 x x = 9 tan 17° x = = 29.44

Example 2 Use the sine ratio to find the value of the variable. Round decimals to the nearest tenth. sin A = opposite hypotenuse 14 25° B sin 25° = x 14 x = 5.9 C x A Multiply both sides by sin 25° = x

Example 3 A rope, staked 20 feet from the base of a building, goes to the roof and forms an angle of 58° with the ground. To the nearest tenth of a foot, how long is the rope? Multiply both sides by x. 58° x x cos 58° = 20 Divide both sides by cos 58° cos 58° = adj. hyp. cos 58° = 20 x x = 20 cos 58° ≈ ≈ 37.7 ft 20 ft

Example 4 Find the value of h. Multiply both sides by ° h 24 tan 65° = h Use a calculator to simplify. tan 65° = opp. adj. tan 65° = h 24 h = 51 feet 24 ft

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