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Chapter 5 Section 5.3 Evaluating Trigonometric Functions Hypotenuse, Adjacent and Opposite sides of a Triangle In a right triangle (a triangle with a right angle) the side that does not make up the right angle is called the hypotenuse. For an angle that is not the right angle the other two sides are names in relation to it. The opposite side is a side that makes up the right angle that is across from . The adjacent side is the side that makes up the right angle that also forms the angle . hypotenuse opposite side adjacent side hypotenuse adjacent side opposite side The Trigonometric Ratios hypotenuse For any right triangle if we pick a certain angle we can form six opposite side different ratios of the lengths of the sides. They are the sine, cosine, tangent, cotangent, secant and cosecant (abbreviated adjacent side sin, cos, tan, cot, sec, csc respectively). opposite side opposite side hypotenuse tan sin sec adjacent side hypotenuse adjacent side hypotenuse adjacent side adjacent side csc cos cot opposite side hypotenuse opposite side To find the trigonometric ratios when the lengths of the sides of a right triangle are known is a matter of identifying which lengths represent the hypotenuse, adjacent and opposite sides. In the triangle below the sides are of length 5, 12 and 13. We want to find the six trigonometric ratios for each of its angles and . sin 5 13 cos 12 13 tan 5 12 cot 12 5 sec 13 12 csc 13 5 sin 12 13 cos 5 13 12 tan 12 5 Notice the following are equal: cot 5 12 sec 13 5 csc 13 12 13 5 sin cos sin cos tan cot tan cot sec csc sec csc The angles and are called complementary angles (i.e. they sum up to 90). The “co” in cosine, cotangent and cosecant stands for complementary. They refer to the fact that for complementary angles the complementary trigonometric ratios will be equal. (i.e. sin 𝐴 = cos 90° − 𝐴 ,tan 𝐴 = cot 90° − 𝐴 , etc.) Trigonometric Ratios of Special Angles 45-45-90 Triangles 30-60-90 Triangles If you consider a square where each side is of length 1 then the diagonal is of length 2. If you consider an equilateral triangle where each side is of length 1 then the perpendicular to the other side is of Ratios are: 2 2 : :1 2 2 45 1 x 2 1 45 sin 45 cos 45 1 2 2 2 1 1 1 cot 45 (Ratios are 1 3 : : 1) 2 2 30 x 1 2 2 2 tan 45 length 3 . 2 1 1 1 sec 45 2 1 csc 45 2 1 1 1 x 2 11 x2 2 x2 2 2 60 1 2 2x 3 2 2 sin 60 3 2 cos 60 1 2 tan 60 cot 60 sec 60 csc 60 3 1 3 3 3 2 2 2 3 3 3 1 2 x 2 12 12 2 x 2 14 1 x2 x sin 30 1 2 cos 30 3 2 tan 30 1 3 3 3 cot 30 3 3 4 sec 30 3 2 csc 30 2 2 3 3 3 2 The values of the trigonometric functions will be the same as that of the trigonometric ratios. In particular for the angles 0°,30°,45°,60°,90°. In the picture to the right the first quadrant is shown along with the terminal points on the unit circle. sin t cos t tan t cot t sec t csc t 0 0 1 0 30 1 2 3 2 3 3 45 2 2 2 2 1 3 2 1 2 1 0 60 90 3 - - 1 - 3 2 3 3 2 1 2 2 3 3 2 2 3 3 0 - 1 t 90 , P(0,1) t 60 , P 12 , 3 2 t 45 , P 2 2 , t 30, P 2 2 , 3 2 1 2 t 0 , P(1,0) Reference Angles The reference angle for an angle is the angle made when you drop a line straight down to the x-axis. it is the angle made by the x-axis regardless of what side of it you are on. (Go to closest multiple of 180° and add or subtract.) 225 120 330 60 60 30 45 -300 240 Helps to find the values. Find the six trigonometric − 3 2 ratios for 240° angle. 60 −1 2 − 3 = 2 tan 240° = 3 sin 240° sec 240° = −2 −1 = 2 1 ° cot 240 = 3 −2 ° csc 240 = 3 cos 240° Find the values for a, b, c, and d in the triangles pictures to the right. 45° 24 c To find a: b 60° a 𝑎 0 cos 60 = 24 1 𝑎 = 2 24 𝑎 = 12 d To find c: 𝑏 = 𝑐 2 12 3 = 2 𝑐 cos 450 To find b: 𝑏 = 24 3 𝑏 = 2 24 𝑏 = 12 3 sin 60° To find d: 𝑐= 24 3 2 tan 45° 1= 𝑑 = 𝑏 𝑑 12 3 𝑑 = 12 3