How do you determine the exact values of the six trig function of the angle given (-4,10)?

2 Answers
Jul 2, 2015

Find 6 trig functions.

Explanation:

On the trig unit circle,

tan a = x/y = 4/10 = 0.40 -> a = 21.80 deg

x = a + 90 = 90 + 21.80 = 111.80 deg (Quadrant II)

sin x = sin 111.80 = 0.93
cos x = cos 111.80 = - 0.37

#tan x = 0.93/-0.37 = - 2.50#
#cot x = 1/-2.50 = - 0.40#
#sec = 1/-0.37 = - 2.70#
#csc x = 1/0.93 = 1.07#

Jul 2, 2015

Assuming that the point #(-4, 10)# is on the terminal side of the angle, we can use the definitions on the trigonometric functions. We do not need to find the angle.

Explanation:

If point #(x,y)# is on the terminal side of an angle #theta# in standard position, then the trigonometric functions are defined using #r# which is the distance between #(x,y)# and the origin: #r = sqrt(x^2+y^2)#
In this case #r = sqrt((-4)^2 + (10)^2) = sqrt 116 = sqrt (4*29) = 2sqrt29#

#sin theta = y/r# In this case #sin theta = 10/(2sqrt29) = 5/sqrt29 = (5sqrt29)/29#

#cos theta = x/r#, in this case #cos theta = -4/(2sqrt29) = - 2/sqrt29 = -(2sqrt29)/29#

#tan theta = y/x#. In this case #tan theta = 10/(-4) = -5/2#

the other functions are the reciprocals of these:

#csc theta = r/y = (2sqrt29)/10 = sqrt29/5#

#sec theta = r/x = (2sqrt29)/-4 = -sqrt29/2#

#cot theta = x/y = (-4)/10 = -2/5#.

It is very important to memorize the definitions of the trigonometric functions. (Flash card are a great idea -- make some.)