Question

Solve this trigonometric equation.give all positive values of the angle between 0deg and 360deg that will satisfy.
Give any approximate value to the nearest minute only.

3 sin theta - 4 cos theta = 2

Answer 1

theta = 209,55192°
theta = 76,70829°

Answer 2

Sum or difference of sin(θ) and cos(θ) can be solved in the following way, if you are familiar with the sum and difference formulae for sin(a±b).

Given
3sin(θ)-4cos(θ)=2
for a particular φ and amplitude A, we rewrite the above equation as:
Asin(θ-φ)=2
where A=sqrt(3²+4²)=5
or
sin(θ-&phi)=2/5....(1)
=> θ-&phi=asin(2/5)=23.5782°
or 180-23.5782=156.4218°. ...(2)

Expand (1) to get:
sin(θ)cos(&phi)-cos(θ)sin(&phi)=2/5
=> cos(φ)=3/5, sin(φ)=4/5
=> φ=53.1301° or 306.8699°...(3)

So solve for θ from (2) and (3) gives
θ=23.5782+53.1301=76.7083° or θ=156.4218+53.1301=209.5519°