Verified answer

1) If sin < 0 and tan is > 0, then cos < 0 (Quad III)

use pythagorean theorem to find the missing leg for cos

You know the hypotenuse = 13, and one leg is = 12. what is the other leg?

x^2 = 13^2 - 12^2 = 169 - 144 = 25

x^2 = 25 --> x = 5

(5, 12, 13 is a common pythagorean triple. save yourself some time and learn some of these triples)

cos (theta) = - 5/13

2) sec (theta) = √26 / 5 and sin < 0

sec = 1/cos, and is positive. cos > 0

than means tan < 0

cos (theta) = 5/ √26

we know the hypotenuse , and one leg . what is the missing leg?

x^2 = (√26)^2 - 5^2 = 26 - 25 = 1

x^2 = 1 --> x = 1

sin (theta) = - 1/√26

tan (theta) = sin (theta) / cos (theta) = (-1 / √26) / (5 /√26) = -1/5

1) -5/13

2) -1/5

Draw out the triangle and find the third leg using the Pythagorean Theorem. Then use the quadrant values to determine the sign.

1.) cos(θ) = -√(1 - sin²(θ)) = -√1 - 144/169) = -5/13

2.) sin(θ) = -√(1 - cos²(θ)) = -√(1 - 25/26) = -1/√26 = -√26/26

tan(θ) = sin(θ)/cos(θ) = -√26/26 * √26/5 = -1/5