Getting the exact value without calculator?

Question

If cos theta = -12/13 when pi/2 <= theta <= pi and sin alpha = -3/5 when pi/2 <= delta <= pi, find tan(theta + delta)

 

Thanks for your help.

 

Answer 1

Hi Mario

 

Recall cos θ = adjacent / hypotenuse = -12/13

between π/2 and π, i.e. in the 2nd quadrant.

To determine the sin θ, we need to use the pythagorean theorem a² + b² = c², where a and b are the legs and c the hypotenuse of the right triangle.

In our case

a² + (-12)² = 13²

a² + 144 = 169

a² = 25

a = 5 opposite

Since sin θ = opposite / hypotenuse = 5/13

And tan θ = opposite / adjacent = -5/12

 

Sin δ = opposite / hypotenuse = -3/5    

between π/2 and π, i.e. in the 2nd quadrant.

To determine the sin θ, we need to use the pythagorean theorem a² + b² = c², where a and b are the legs and c the hypotenuse of the right triangle.
In our case
(-3)² + b² = 5²
9 + b² = 25
b² = 14
b = 4  adjacent
Since cos δ = adjacent / hypotenuse = 4/5
And tan δ = opposite / adjacent = -3/4

 

 

The tan of the sum of two angles is

                     1−tan α tan β

tan(α+β)= -----------------------
                     tan α+tan β

 

1 - (-5/12)(-3/4)

----------------------

 (-5/12) + (-3/4)

      1 - 15/48

--------------------

(-5/12) + (-9/12)

 

 33/48

--------

-14/12

 

33/48 • -12/14 = -33/56