Question

if 4cos theta + 3 sin theta = 5, find the value of tan theta.

Answer 1

3 sin theta = 5-4 cos theta
divide the above by cos theta
3 tan theta = 5 sec theta - 4
3 tan theta + 4= 5 sec theta
squaring on both sides we get
9 tan²theta + 16 + 24 tan theta = 25 sec² theta
9 tan²theta + 16 + 24 tan theta = 25 (1+tan²theta)
25 + 25 tan² theta - 9 tan² theta - 16 - 24 tan theta = 0
16 tan² theta - 24 tan theta + 9 = 0
Comparing the above equation with the quadratic eqn
we have a = 16, b= -24 and c = 9
Now substituting these values in the roots of the quadratic eqn
we get tan theta = 3/4

Answer 2

3 sin theta = 5-4 cos theta

divide by cos theta

3 tan theta = 5 sec theta - 4

3 tan theta + 4= 5 sec theta

square both sides

9 tan²theta + 16 + 24 tan theta = 25 sec² theta

9 tan²theta + 16 + 24 tan theta = 25 (1+tan²theta)

25 + 25 tan² theta - 9 tan² theta - 16 - 24 tan theta = 0

16 tan² theta - 24 tan theta + 9 = 0

Comparing the above equation with the quadratic equation

we have a = 16, b= -24 and c = 9

Now substituting these values in the roots of the quadratic eqn

we get tan theta = 3/4