Find sin(t), sec(t), csc(t), tan(t), and cot(t)

Question

If cos(t)=-2/5, and t in in quadrant III, find the exact values of sin(t), sec(t), csc(t), tan(t), and cot(t).

Answer 1

Start by sketching a set of axes and drawing a right triangle in the third (bottom left) quadrant; it doesn't have to be perfectly to scale, as long as the orientation is correct. Draw the hypotenuse out from the origin and then connect it up to the x-axis making a right triangle. Angle t is the one closest to the origin.

Cosine is adjacent/hypotenuse so label the sides of the triangle based on the value given. The hypotenuse is always positive, which means the adjacent side is -2 (which makes sense because the triangle is oriented along the negative part of the x-axis). Then you need to find the third side using the pythagorean theorem, and that side would be negative too (because the triangle is also in negative y-axis land).

Once you have all three sides of the triangle, it should be simple to figure out the rest of the trig functions.

I hope this helps, let me know if you have more questions.

Answer 2

cos(t)^2 + sin(t)^2 = 1

so

(-2/5)^2 + sin(t)^2 = 1

solve for sin(t) - since in QIII, it will be negative

sec(t) = 1/cos(t)

csc(t) = 1/sin(t)

tan(t) = sin(t)/cos(t)

cot(t) = 1/tan(t)