How do I find individual values of $\sin(\varphi)$ and $\cos(\varphi)$ from $x = a\sin^{2}(\varphi) + b\cos^{2}(\varphi)$?

Question

If $x = a\sin^2\phi+ b\cos^2\phi$, express $\sin$ and $\cos$ in terms of $x$ ( $a$ and $b$ are real constants)

I know how to find values of $T$ ratios in equations like $x = a\sin^2t$ or $x = b\cos^2t$ but how do I find the values of $\sin(t)$ and $\cos(t)$ in expressions like $x = \sin^2t + 5\cos^2t$ or $x = a\sin t + b\cos t$?

Answer 1

HINT

I would recommend you to proceed as follows:

\begin{align*} x & = a\sin^{2}(\varphi) + b\cos^{2}(\varphi)\\\\ & = (a\sin^{2}(\varphi) + a\cos^{2}(\varphi)) + (b - a)\cos^{2}(\varphi)\\\\ & = a + (b - a)\cos^{2}(\varphi) \end{align*}

Can you take it from here?

Answer 2

1. $$x=a\sin^2\phi + b\cos^2\phi$$

2. $$x=a\sin^2\phi+b-b\sin^2\phi$$

3. $$x=b+(a-b)\sin^2\phi$$

4. $$\sin\phi=\pm\sqrt\frac{x-b}{a-b}$$

$$\cos\phi=\pm\sqrt\frac{a-x}{a-b}$$

Hope this helps!!