Sine squared function solving for the variable a degrees

Question

Find degree a:

$$\sin^{2}(a-20)-\sin^{2}(a+20) = 0.1714$$

Everything is in degrees.

How do I solve this for a?

Answer 1

a nice identity: $$\sin^2(x)-\sin^2(y)=\sin(x+y)\, \sin(x-y)$$

So, $$\sin(2a)\sin(-40)=0.1714$$ from there you can find $2a$ by a calculator.

Answer 2

...or you can use the conjugate rule:

$$\sin^{2}(a-20)-\sin^{2}(a+20)=(\sin(a-20)+\sin(a+20))(\sin(a-20)-\sin(a+20))= -2\sin a\cos20\cdot 2\cos a\sin20=-2\sin20\cos20\sin(2a)=-\sin(40)\sin(2a)$$

Answer 3

Hint: make use of the identities:

$$\sin{\alpha \pm \beta} = \sin{\alpha} \cos{\beta} \pm \cos{\alpha} \sin{\beta},$$ with $\alpha = a$ and $\beta = 20º$. When you substitute this back in your equation, the $\sin^2 \cos^2$ operators will vanish so you will end up for a single equation for $\sin{a}\cos{a} = \pm \sin{a} \, \sqrt{1-\sin^2{a}}$.

Hope this helps.

Edit: a much more nicer answer is provided by the user @mesel.

Cheers!