How do you use the unit circle to find the values of the six trigonometric functions for 150 degrees?

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Answer 1 · 2018-06-17T03:59:08

$cos 150 = - \frac{\sqrt{3}}{2}$ $cos150=-sqrt3/2$

$sin 150 = \frac{1}{2}$ $sin150=1/2$

$tan 150 = - \frac{\sqrt{3}}{3}$ $tan150=-sqrt3/3$

$cot 150 = - \sqrt{3}$ $cot150=-sqrt3$

$sec 150 = \frac{- 2 \sqrt{3}}{3}$ $sec150=(-2sqrt3)/3$

$csc 150 = 2$ $csc150=2$

We know the reference angle is $30^{\circ}$ $30^@$ , and from the unit circle, we know the coordinates for $30^{\circ}$ $30^@$ are

$(\frac{\sqrt{3}}{2}, \frac{1}{2})$ $(sqrt3/2,1/2)$

Our angle, $150^{\circ}$ $150^@$ is in the second quadrant, where cosine is negative and sine is positive. Unit circle coordinates are given by

$(cos θ, sin θ)$ $(costheta, sintheta)$

This means the coordinates for $150^{\circ}$ $150^@$ are

$(- \frac{\sqrt{3}}{2}, \frac{1}{2})$ $(-sqrt3/2,1/2)$

We know:

$cos 150 = - \frac{\sqrt{3}}{2}$ $color(blue)(cos150=-sqrt3/2)$

$sin 150 = \frac{1}{2}$ $color(darkblue)(sin150=1/2)$

$tan θ = \frac{sin θ}{cos θ}$ $color(lime)(tantheta)=color(darkblue)(sintheta)/color(blue)(costheta)$

And from our definitions of trig functions:

$cot θ = \frac{1}{tan θ}$ $color(red)(cottheta)=1/color(lime)(tantheta)$

$sec θ = \frac{1}{cos θ}$ $color(darkviolet)(sectheta)=1/color(blue)(costheta)$

$csc θ = \frac{1}{sin θ}$ $color(orange)(csctheta)=1/color(darkblue)(sintheta)$

After plugging in the appropriate values (and rationalizing the denominator when necessary), we get

$tan 150 = - \frac{\sqrt{3}}{3}$ $color(lime)(tan150=-sqrt3/3)$

$cot 150 = - \sqrt{3}$ $color(red)(cot150=-sqrt3)$

$sec 150 = \frac{- 2 \sqrt{3}}{3}$ $color(darkviolet)(sec150=(-2sqrt3)/3)$

$csc 150 = 2$ $color(orange)(csc150=2)$

Hope this helps!