Solving a*sin($\theta$) + b*cos($\theta$) - c*cos($\theta$) = d

Question

Given $a, b, c, d$. How do I get the theta?

$a \cdot \sin(\theta) + b \cdot \cos(\theta) - c \cdot \cos(\theta) = d$

Answer 1

The first method is to isolate $(b-c)\cos(\theta)$ and square the equation $$(b-c)\cos(\theta)=d-a\sin(\theta)$$ so we get $$(b-c)^2\cos^2(\theta)=d^2+a^2\sin^2(\theta)-2ad\sin(\theta)$$ with $\cos^2(\theta)=1-\sin^2(\theta)$ you will get an quadratic equation to solve.