Use fundamental identities to find the values of the trigonometric functions for the given conditions.

Question

Use fundamental identities to find the values of the trigonometric functions for the given conditions.

sin 𝜃 = −12/13 and sec 𝜃 > 0

sin 𝜃	=
cos 𝜃	=
tan 𝜃	=
csc 𝜃	=
sec 𝜃	=
cot 𝜃	=

Answer 1

Determine what quadrant the triangle in question is in

1. Since secant is the reciprocal of cosine, consider the quadrants cosine is positive - that would be Q1 and Q4
2. Consider what quadrant sine is negative - that would be Q3 and Q4
3. So our triangle must be in Q4

Now, sketch the triangle based on the fact that sine is opposite/hypotenuse so the opposite side would be -12 (matches the Q4 condition) and the hypotenuse must be 13:

Use the Pythagorean Theorem to solve for "x"

x² + (-12)² = 13²

x² + 144 = 169

x² = 25 or x = 5

Now, write the trig ratios using the adjacent side = 5, the opposite side = -12 and the hypotenuse = 13

For example: cos(θ) = adjacent/hypotenuse = 5/13

Answer 2

If sec > 0 then cos = 1/sec > 0. Since cos is positive and sin is negative functions are in quadrant IV

sin^2 + cos^2 = 1

(-12/13)^2 + cos^2 = 1

144/169 + cos^2 = 1

cos^2 = 25/169

cos = 5/13 (since cos is positive)

tan = sin/cos = -12/5

cot = 1/tan = -5/12

sec = 1/cos = 13/5

csc = 1/sin = -13/12