Section 5.3 Evaluating Trigonometric Functions 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 adjacent side opposite side  The Trigonometric Ratios For any right triangle if we pick a certain angle  we can form six different ratios of the lengths of the sides. They are the sine, cosine, tangent, cotangent, secant and cosecant (abbreviated sin, cos, tan, cot, sec, csc respectively).  hypotenuse opposite side adjacent side

Notice the following are equal: 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 . 5 12 13   sin  cos  tan  cot  sec  csc  sin  cos  tan  cot  sec  csc  Notice the following are equal: 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 If you consider a square where each side is of length 1 then the diagonal is of length 2 . 30-60-90 Triangles If you consider an equilateral triangle where each side is of length 1 then the perpendicular to the other side is of length 3 2 . (Ratios are 1 2 : 3 2 :1) Ratios are: 2 2 : 2 2 :1 1 45 1 30 60 sin 60 cos 60 tan 60 cot 60 sec 60 csc 60 sin 30 cos 30 tan 30 cot 30 sec 30 csc 30 sin 45 cos 45 tan 45 cot 45 sec 45 csc 45

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. 1 - 2

Helps to find the values. 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.) 120 60 225 45 -300 60 330 30 240 60 sin 240 ° = − 3 2 cos 240 ° = −1 2 Helps to find the values. Find the six trigonometric ratios for 240 ° angle. −1 2 cot 240 ° = 1 3 tan 240 ° = 3 − 3 2 csc 240 ° = −2 3 sec 240 ° =−2

Find the values for a, b, c, and d in the triangles pictures to the right. 45 ° 24 c To find b: sin 60 ° = 𝑏 24 3 2 = 𝑏 24 𝑏=12 3 To find a: cos 60 0 = 𝑎 24 1 2 = 𝑎 24 𝑎=12 b 60 ° a d To find c: cos 45 0 = 𝑏 𝑐 2 2 = 12 3 𝑐 To find d: tan 45 ° = 𝑑 𝑏 1= 𝑑 12 3 𝑐= 24 3 2 𝑑=12 3