Precalculus/Trigonometric Functions of Sine, Cosine, and Tangent with given parameters? for my precalculus class I was given an assignment for extra credit however it is some material that I have yet to cover or learn as far as sine, cosine, and tangent go. Below is the prompt that I was given:
If sin α = 3/13 where 0 < α < π/2 and cos β = -2/9 where π < β < 3π/2, determine the following:


*

*sin(α - β)

*cos(β - α)

*tan(2β)

*cos(4β)

*cot(α + β)


I am not quite sure of the first step to take on these. I wasn't able to find much online either due to my search terms or what not. I hate to upload such a blank question however I am at a loss for how to approach these questions...
 A: They require you to use the identities: $$\sin(\alpha\pm \beta)=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta$$ $$\cos (\alpha\pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta$$
From these one can infer the double-angle formulae:
$$\sin 2\alpha = 2\sin \alpha\cos \alpha$$ $$\cos 2\alpha = \cos^2 \alpha -\sin^2 \alpha = 2\cos^2 \alpha -1 =1-2\sin^2 \alpha$$ $$\tan 2\alpha = \frac {2\tan \alpha}{1-\tan^2\alpha}$$
To find the value of $\sin\beta$ or $\cos\alpha$, draw a triangle with the lengths of two sides given by the fractions $\frac 3 {13}$ or $\frac {-2} 9$. Use Pythagoras' Theorem to find the length of the third side, and infer the value of $\sin\beta$ or $\cos\alpha$. 

You could use the CAST diagram to work out whether $\sin\beta$ and $\cos\alpha$ are positive or negative, by looking at where the angles $\alpha$ and $\beta$ lie in a circle.
A: Applying the expressions
$$\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta$$
$$\cos(\beta-\alpha)=\cos\alpha\cos\beta+\sin\alpha\sin\beta$$
$$\tan(2\beta)=\frac{2\sin\beta\cos\beta}{\cos^2\beta-\sin^2\beta}$$
$$\cos(4\beta)=\cos^2(2\beta)-\sin^2(2\beta)=(\cos^2\beta-\sin^2\beta)^2-(2\sin\beta\cos\beta)^2$$
$$\cot(\alpha+\beta)=\frac{\cos\alpha\cos\beta-\sin\alpha\sin\beta}{\sin\alpha\cos\beta-\cos\alpha\sin\beta}$$
where you can get
$$\cos\alpha=\sqrt{1-\sin^2\alpha}$$
$$\sin\beta=-\sqrt{1-\cos^2\beta}$$
because cosine is positive in the first quadrant and sine negative in the third.
A: If you want calculate this, you can use this site: http://www.milefoot.com/math/trig/22anglesumidentities.htm.
With those formulas your work is just analyze the signals of the functions in these intervals, can you do it or you don't know how analyze the signals ?
