14-3 Right Triangle Trig Hubarth Algebra II

The trigonometric ratios for a right triangle: A B C a b c

Ex. 1 Real-World Connection A tourist visiting Washington D.C. is seated on the grass at point A and is looking up at the top of the Washington Monument. The angle of her line of sight with the ground is 27°. Given that sin 27° 0.45, cos 27° 0.89, and tan 27° 0.51, find her approximate distance AC from the base of the monument. The distance of the visitor from the monument is approximately 1088 feet. tan 27° = Substitute. 555 AC 0.51 = Use a calculator in degree mode. 555 AC tan 27° = Definition of tan height of monument distance from monument

Ex. 2 Using a Right Triangle to Find Ratios Step 1: Draw a diagram.Step 2: Use the Pythagorean Theorem to find p. r 2 = p 2 + q = p = p = p 2 24 = p Step 3: Calculate the ratios. In PQR, R is a right angle and cos P =. Find sin P, tan P, and cos Q in fraction and in decimal form. 7 25

A man 6 feet tall is standing 50 feet from a tree. When he looks at the top of the tree, the angle of elevation is 42°. Find the height of the tree to the nearest foot. In the right triangle, the length of the leg adjacent to the 42° angle is 50 ft. The length of the leg opposite the 42° angle is unknown. You need to find the length of the leg opposite the 42° angle. Use the tangent ratio. The height of the tree is approximately or 51 ft. x = 50 tan 42° Solve for x. tan 42° = Definition of tan x Use a calculator in degree mode. Ex. 3 Real-World Connection

In KMN, N is a right angle, m = 7, and n = 25. Find m K to the nearest tenth of a degree. Step 1: Draw a diagram.Step 2: Use a cosine ratio. cos K = = 0.28 m K = cos – °Use a calculator Since K is acute, the other solutions of cos – do not apply. To the nearest tenth of a degree, m K is 73.7°. Ex. 4 Finding Angle Measures

Ex. 5 Real-World Connection A straight road that goes up a hill is 800 feet higher at the top than at the bottom. The horizontal distance covered is 6515 feet. To the nearest degree, what angle does the road make with level ground? You know the length of the leg opposite of the angle you need to find. You know the length of the hypotenuse. So, use the tangent ratio. The angle the road makes with level ground is approximately 7°. Let = the measure of the angle of the slope of the road. tan = = tan –1 Use the inverse of the tangent function Use a calculator.

Practice 1. Find the length of AB, in the figure D E F A B C A B C