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

$cos150=-sqrt3/2$

$sin150=1/2$

$tan150=-sqrt3/3$

$cot150=-sqrt3$

$sec150=(-2sqrt3)/3$

$csc150=2$

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

$(sqrt3/2,1/2)$

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

$(costheta, sintheta)$

This means the coordinates for $150^@$ are

$(-sqrt3/2,1/2)$

We know:

$color(blue)(cos150=-sqrt3/2)$

$color(darkblue)(sin150=1/2)$

$color(lime)(tantheta)=color(darkblue)(sintheta)/color(blue)(costheta)$

And from our definitions of trig functions:

$color(red)(cottheta)=1/color(lime)(tantheta)$

$color(darkviolet)(sectheta)=1/color(blue)(costheta)$

$color(orange)(csctheta)=1/color(darkblue)(sintheta)$

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

$color(lime)(tan150=-sqrt3/3)$

$color(red)(cot150=-sqrt3)$

$color(darkviolet)(sec150=(-2sqrt3)/3)$

$color(orange)(csc150=2)$

Hope this helps!