Find the value of the other five trigonometry functions. Also find the reference angle and use it to determine the least positive value of θ to the nearest 0.1°. cosθ = -5/13, tanθ < 0

Answer 1

First, we must remember what the 6 functions are and how to find them:

sin = (opp.)/(hyp.)

cos = (adj.)/(hyp.)

tan = sin/cos = (opp.)/(adj.)

csc = 1/sin = (hyp.)/(opp.)

sec = 1/cos = (hyp.)/(adj.)

cot = 1/tan = cos/sin = (adj.)/(opp.)

hyp. = hypotenuse; is always assumed positive

opp. = opposite side

adj. = adjacent

Second, we can determine what the adjacent side and hypotenuse is by the above relation: cos = (adj.)/(hyp.)

cos∅ = (-5)/13 ---> adj. = -5

hyp. = 13

Third, we can determine what the opp. is by using the Pythagorean Theorem (hyp.)² = (adj.)² + (opp.)²:

13² = (-5)² + (opp.)²

169 = 25 + (opp.)²

169 - 25 = (opp.)²

144 = (opp.)²

opp. = 12

Finally, we know that cosine and tangent are both negative only in Quadrant II, so ∅ must be between 90° and 180° and the reference angle is the angle between the 180° line and the line created by the information given. You can solve the other 5 functions with the information you have found.

Hope this helps!