Question

Evaluate (if possible) the six trigonometric functions of the angle (0).
-5pi/6

Answer 1

Answer/Step-by-step explanation:

Use a Unit Circle, you may have access to (hopefully) or have been asked to memorize.

To find -5pi/6 count clockwise the sixths to 5. Or add:

-5pi/6 + 2pi

= -5pi/6 + 12pi/6

= 7pi/6

-5pi/6 is the same as 7pi/6 on the Unit Circle.

Use the coordinates at that spot

(-root3/2 , -1/2)

to find the six trig ratios.

The Unit Circle is a big, giant, Answer Key. It has all the trig answers on it all in one place. It looks like a scary math pizza, but really its all the answers all in one place.

Once you find your angle, the sine is the y-coordinate.

sin theta = -1/2

The cosine is the x-coordinate.

cos theta = -root3/2

Tangent theta is the sine over the cosine, sin/cos,

= y/x

= (-1/2) / (-rt3/2)

Use Keep-Change-Flip to simplify this.

see image.

= root3/3

Sec is cos flipped over. Csc is sin flipped over. Cot is tan flipped over (the math word is reciprocal)

Yes, just flip over sin to get csc:

sintheta = -1/2

csctheta = - 2/1 = -2

Flip over cos to get sec:

costheta = -rt3/2

sectheta = 2/-rt3

Then rationalize, see image.

= -2rt3/3

Flip over tan to get cot.

tantheta = 1/rt3

Use a form that's easy to flip.

cottheta = rt3/1 = rt3

see image.

Answer 2

• sin(-5π/6) = -1/2
• cos(-5π/6) = -√3/2
• tan(-5π/6) = √3/3
• csc(-5π/6) = -2
• sec(-5π/6) = -2√3/3
• cot(-5π/6) = √3

Explanation:

You want the 6 trig functions of the angle -5π/6.

Quadrant

The angle is in the 3rd quadrant, so all trig functions are negative except the tangent and cotangent. The reference angle is -5π/6 +π = π/6.

Trig functions

Since you have memorized the values of the trig functions for multiples of π/6, you know ...

sin(-5π/6) = -1/2

cos(-5π/6) = -√3/2

tan(-5π/6) = sin/cos = √3/3

csc(-5π/6) = 1/sin = -2

sec(-5π/6) = 1/cos = -2√3/3

cot(-5π/6) = 1/tan = √3