A point P(x, y) is shown on the unit circle U corresponding to a real number t.

Question

A point P(x, y) is shown on the unit circle U corresponding to a real number t. Find the values of the trigonometric functions at t : sin t , cos t , tan t, csc t, sec t, cot t . Assume a = − 12/37 , b = 35/37 .

Answer 1

The point on the unit circle is (cos,sin)

To find tan x = sin x/cos x

So the 6 trig functions are the following:

sin x = 35/37

cos x = −12/37

tan x = (35/37)/(−12/37) = -35/12

To find the rest, these are the reciprocal functions. So take the reciprocal of them.

csc x = 37/35

sec x = -37/12

cot x = -12/35

Answer 2

In this case, it is useful to think about the geometric representations of sine and cosine on the unit circle. On the unit circle, the sine is the y value of the point t and the cos is the x-value of the point t, while tan is the slope at the point t. In this case, the y value given is 35/37, which is sin(t). The x - value given, -12/37, is cos(t). The tan is the slope or alternatively, the y value/ x value = sin t / cos t = 35/37 / -12/37 = -35/12. The cosecant (csc t) is 1/sin t = 1/(35/37) = 37/35. The secant (sec t) is 1/cos t = 1/(-12/37) = -37/12. Finally the cotangent (cot t) is 1/tan t = 1/(-35/12) = -12/35.