SOLUTION: Find the exact value of tan (a + b) if cos a = -3/5, sin b = 5/13, 90° < a < 180°, and 90° < b < 180°. Do not round.

Algebra ->  Trigonometry-basics -> SOLUTION: Find the exact value of tan (a + b) if cos a = -3/5, sin b = 5/13, 90° < a < 180°, and 90° < b < 180°. Do not round.       Log On


   

Question 1014959: Find the exact value of tan (a + b) if cos a = -3/5, sin b = 5/13, 90° < a < 180°, and 90° < b < 180°. Do not round.

Found 2 solutions by robertb, ikleyn:
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
sec a = -5/3 ==> 1 + tan^2 a = 25/9 ==> tan^2 a = 16/9 ==>tan a = -4/3 (Angle a is in the 2nd quadrant because 90° < a < 180°.)
sin b = 5/13 ==> 1+ cot^2 b = 169/25 ==> cot^2 b = 144/25 ==> cot b = -12/5 (Angle b also in the 2nd quadrant because 90° < b < 180°.)
==> tan b = -5/12.
==>

Answer by ikleyn(52800) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find the exact value of tan (a + b) if cos a = -3/5, sin b = 5/13, 90° < a < 180°, and 90° < b < 180°. Do not round.
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1.  Use the addition formula for tangent:  tan(a + b) = %28tan%28a%29%2Btan%28b%29%29%2F%281-tan%28a%29%2Atan%28b%29%29

    (see the lesson Addition and subtraction formulas in this site).


2.  sin(a) = sqrt%281-cos%5E2%28a%29%29 = sqrt%281-%28%28-3%29%2F5%29%5E2%29 = . . . = 4%2F5.

     Notice that the sign of the square root is chosen accordingly with the location of the angle "a" in the Quadrant 2.


3.  cos(b) = -sqrt%281-sin%5E2%28b%29%29 = -sqrt%281-%285%2F13%29%5E2%29 = . . .  = -12%2F13.

     Notice that the sign of the square root is chosen accordingly with the location of the angle "b" in the Quadrant 2.


4.  Now,  tan(a) = sin%28a%29%2Fcos%28a%29 = 4%2F5 : %28-3%2F5%29 = -4%2F3   and

          tan(b) = sin%28b%29%2Fcos%28b%29 = 5%2F13%29 : %28-12%2F13%29 = -5%2F12.


5.  Finally,  tan(a+b) = %28tan%28a%29%2Btan%28b%29%29%2F%281-tan%28a%29%2Atan%28b%29%29 = %28%28-4%2F3%29%2B%28-5%2F12%29%29%2F%281-%28-4%2F3%29%2A%28-5%2F12%29%29 = %28-21%2F12%29%2F%281+-+%2820%2F36%29%29 = %28-21%2F12%29%2F%2816%2F36%29 = -63%2F16.