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How To Remember Special Values of Sine and Cosine The following is a special table for remembering the special exact values of the sine and cosine functions in Quadrant I. The key to the following table is just knowing a few simple patterns. The first is to know the important angles in Quadrant I: These are (in degrees) 0◦ , 30◦ , 45◦ , 60◦ , 90◦ (note the quadrantal angles 0◦ and 90◦ have been included). The second is to√recognize the special n values of the sine and cosine functions all have the form as follows: 2 θ sin θ √ 0 = 0◦ π = 30◦ 6 π = 45◦ 4 π = 60◦ 3 π = 90◦ 2 0 2 √ 1 2 √ 2 2 √ 3 2 √ 4 2 cos θ = 0 =0 2 = 1 2 1 =√ 2 = 2 =1 2 √ 4 2 √ 3 2 √ 2 2 √ 1 2 √ 0 2 = 2 =1 2 1 =√ 2 = 1 2 = 0 =0 2 In order to reproduce the above table, you also need to remember that sin θ increases from 0 to 1 in Quadrant I, while cos θ decreases from 1 to 0 in Quadrant I. There are several ways to keep them straight: One way is you can remember the graphs of the sine and cosine function and thereby recall that sin 0 = 0, cos 0 = 1. Another way is to recall the unit circle picture of the trigonometric functions. Recall that on the unit circle, at angle θ, (x, y) = (cos θ, sin θ). As θ = 0 is the positive x axis—which hits the unit circle at (1, 0)—we have (1, 0) = (x, y) = (cos 0, sin 0), cos 0 = 1, sin 0 = 0. The values of angles outside Quadrant I can be computed using reference angles, and the values of the other trigonometric functions can be computed using the reciprocal and quotient identities. Example: Compute sec π4 and tan π3 . √ 3 π √ √ sin 3 2 1 π π 1 = 3. = = = 2, tan = sec = π π 1 3 cos 3 4 cos 4 √1 2 2