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

Question

14242 views around the world

You can reuse this answer
Creative Commons License

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!