if tana=-(√11)/5 and sinb= (2√10)/7 with pi/2 < a,b < pi find the value of cos(a+b)

Question

the correct answer is (15-2√110)/42

confused on how exactly to set up solving for this

Answer 1

Remember that both angles lie in the 2nd quadrant, sin a and sin b will be positive. cos a, cos b, tan a, and tan b will all be negative,

When we are given the value of any trig function, we are given two of the three values we need to find the third value. You can think of the two values as two sides of a right triangle and use the Pythagorean theorem to find the missing value or use the x, y, r, definitions of the trig functions and use x² + y² = r². These are both exactly the same. Either think of sin θ as the side opposite/the hypotenuse or as y/r. In my solution, I am going to use x, y, and r,

We were given the tan a = -√11/5. Therefor y = √11 and x = -5. Using x² + y² = r², r = 6. sin a = √11/6 and cos a = -5/6. In the same manner given sin b = 2√10/7, cos b = -3/7. (We don't need tan b)

cos(a + b) = cos a cos b - sin a sin b

= (-5/6)(-3/7) - (√11/6)(2√10/7)

=15/42 - 2√110/42

= (15 - √110)/42 Using the correct signs of the angles for the quadrant the angles is in is important. In this case, neglecting the signs would not affect the answer, but that is just a matter of luck not accuracy.

Answer 2

Use the formula

cos(a+b) = cos(a)cos(b) - sin(a)sin(b) (1)

We are given sin(b) = 2√10/7, and need to compute sin(a), cos(a), cos(b):

cos(b) = √(1 - sin²(b)) = √(1 - 40/49) = √(9/49) = 3/7

To calculate sin(a) from tan(a) do the following:

tan(a) = sin(a)/cos(a)

tan²(a) = sin²(a)/cos²(a) = sin²(a)/(1 - sin²(a))

tan²(a)(1 - sin²(a)) = sin²(a)

tan²(a) - tan²(a)sin²(a) = sin²(a)

sin²(a) = tan²(a) / (tan²(a) + 1)

Substitute tan(a) = -√11/5:

sin²(a) = (11/25) / (11/25 + 1) = 11/36

sin(a) = √11/6

cos²(a) = 1 - sin²(a) = 1 - 11/36 = 25/36

cos(a) = 5/6

Substitute the computed quantities into (1)

cos(a+b) = (5/6)(3/7) -(√11/6)(2√10/7) = (15 - 2√110)/42