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Solving Right Triangles Use trigonometric ratios to find angle measures in right triangles and to solve real-world problems. Inverse Functions In Algebra, when we have a function f(x) such as y 2 x 3, we can find a function 1 called the inverse function f x by exchanging places with the x and the y. In this case, if we exchange places with the x and the y, we will get the following function: x 2y 3 or change it to x 3 which would be the inverse y function. 2 Unit J 2 Inverse Functions In the previous lesson, we learned that the trig function of sin 30º = 0.5. We want a function that will tell us that an angle that has a sine of 0.5 would be 30º. This is written as sin-1 (0.5) = 30º and is called an inverse function. For each of the three trigonometric functions, there is an inverse function. Unit J 3 Inverse Functions O To find the inverse function for sin x think of H O the angle as the x and the as the y. H Now change places with the x and the y: 1 O sin x H To access the inverse trig functions on your calculator use 2ND and then SIN , COS , or TAN . Unit J 4 Key Concepts: Get Organized C 5 4 A Trig Function sine sin A = cosine cos A = 3 5 4 5 tan A = 3 4 tangent 3 B Inverse Trig Function 3 5 cos-1 54 3 -1 tan 4 sin-1 Unit J = m∠A = m∠A = m∠A 5 Example of Inverse Functions C Use the given sides to find the measure of angle A. You could use the sine, cosine or 4 tangent function to find the answer. sin A = ( 54 ) use: sin-1 ( 54 ) = 53º cos A = ( 35 ) use: cos-1 ( 35 ) = 53º B 5 A 3 tan A = ( 34 ) use: tan-1 ( 34 ) = 53º No matter which function you use, you will get 53º. Unit J 6 Inverse Functions When you know the degrees of the angle and you need to find the missing side, you will use one of the functions: or. A O O cos x sin x tan x H H A When you know the side lengths of the triangle and you need to find a missing angle, we will use one of the inverse functions: 1 O 1 A 1 O tan x cos x sin x A H H Unit J 7 Lesson Quiz Use your calculator to find each angle measure to the nearest degree. 1. cos-1(0.97) 14° 2. tan-1(2) 63° 3. sin-1(0.59) 36° Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 4. 5. AC 0.63; BC 2.37; mB = 15° Unit J DF 5.7; mD 68°; mF 22° 8