Section 5-5: Solving Right Triangles Your calculator can find sine, cosine, and tangent in degrees and radians. sin cos tan   We will use their definition to find secant, cosecant, and cotangent. csc = 1 𝑠𝑖𝑛 sec = 1 𝑐𝑜𝑠 cot = 1 𝑡𝑎𝑛 When using the calculator, first you must determine if the angle is in DEGREES or RADIANS. Put your calculator in the appropriate MODE. Example 1: Use the calculator to find the following. a) cot 45° b) csc 30° c) sec 22.5° INVERSE OF THE SINE, COSINE AND TANGENT FUNCTIONS How do we solve : sin 𝜃 = -.5 1 tan 45° 1 sin 30° 1 cos 22.5° = 1 = 2 = 1.0823922 𝜃 = sin −1 (−.5) 𝜃 = -30°

That means, sin-1 and sin will cancel each other out, as well as cos and cos-1, or tan and tan-1. We go backward on the calculator to find the angle in degrees or radians using the   sin-1 called arcsine cos-1 called arccosine tan-1 called arctangent Arcsine, arccosine and arctangent are ____________not functions. When going backward, the best method for understanding your key sequence on the calculator is to solve algebraically for 𝜽 first, then type into the calculator. Example 2: Find in degrees. a) cos 𝜃 = .5 b) sin 𝜃 = .7216 c) tan 𝜃 = 1.1256 relations 𝜃= cos −1 (.5) 𝜃 = sin −1 (.7216) 𝜃 = tan −1 (1.1256) 𝜃 = 60° 𝜃 = 46.18673788 𝜃 = 48.38162965 𝜃 ≈ 46.2° 𝜃 ≈48.4°

Example 1: If r = 14 and s = 8, find S. You want to find the measure of an acute angle in a right triangle. You know the side opposite the angle and the hypotenuse. sin S= 𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 sin S= 8 14 S= 𝑠𝑖𝑛 −1 4 7 Angle S is about 34.8°. S = 34.8499

Trigonometry can be used to find the angle of ______________ or the angle of depression. Example 2: HOUSEHOLD The camera for a baby monitor is set up on a shelf in a child’s room, and it is angled so that it captures the image of the center of the baby’s crib. The shelf is about 3 feet higher than the crib, and its horizontal distance from the crib is about 7 feet. What is the angle of depression of the light? The angle of depression and the angle of elevation are equal in measure because they are alternate interior angles. tan 𝜃= 𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 tan 𝜃= 3 7 𝜃= 𝑡𝑎𝑛 −1 3 7 The angle of depression should be about 23.2°. 𝜃=23.1986

You can use trigonometric functions and inverse relations to solve right triangles. To solve a right triangle means to find __________________________ and ____________. all the measures of its sides the angles *** Whenever possible, use the measures given in the problem to find the unknown measures. Example 3: Solve each triangle described, given the triangle at the right.   A. B = 42°, b = 4.5 Find A : A + 42 + 90= 180 A = 48° Find a : tan 48°= 𝑎 4.5 Find c : cos 48°= 4.5 𝑐 4.5 tan 48°=𝑎 𝑐 cos 48°=4.5 a = 4.9978 𝑐= 4.5 cos 48° 𝑎 ≈5 c = 6.7251 𝑐 ≈6.7

B. b = 18, c = 52 Find a : a2 + (18)2 = (52)2 a2 + 324 = 2704 a2 = 2380 a = 2 595 𝑎 ≈48.7852 𝑎 ≈48.8 Find A : sin 𝐴= 48.8 52 A = 𝑠𝑖𝑛 −1 48.8 52 A = 69.7948 A = 69.8° Find B : 69.8°+𝐵+90°=180° B = 20.2°