# Calculate sin (alpha - beta) and cos (alpha + beta)

Trigonometry equalities, inequalities and expressions - sin, cos, tan, cot

### Calculate sin (alpha - beta) and cos (alpha + beta)

Calculate sin (alpha - beta) and cos (alpha + beta)

IF:

cos alpha = - 5/13
sin beta = 12/13

pi/2 < alpha < pi, pi/2 < beta < pi
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### Re: Calculate sin (alpha - beta) and cos (alpha + beta)

I thought it was well known that
$$cos(\alpha+ \beta)= cos(\alpha)cos(\beta)- sin(\alpha)sin(\beta)$$ and
$$sin(\alpha+ \beta)= sin(\alpha)cos(beta)+ sin(\beta)cos(\alpha)$$
(https://brownmath.com/twt/sumdiff.htm)

Now replace $$\beta$$ with $$-\beta$$ and use the fact that cos(-x)= cos(x) and sin(-x)= -sin(x).
Guest

### Re: Calculate sin (alpha - beta) and cos (alpha + beta)

$$Cos(\alpha)=\frac{-5}{13}\Rightarrow Sin^{2}(\alpha)+ Cos^{2}(\alpha)=1 \Rightarrow Sin(\alpha)=\frac{12}{13}$$
$$Sin(\beta)=\frac{12}{13}\Rightarrow Sin^{2}(\beta)+ Cos^{2}(\beta)=1 \Rightarrow Cos(\beta)=\frac{5}{13}$$
$$Sin(\alpha-\beta)=Sin(\alpha)Cos(\beta)-Cos(\alpha)Sin(\beta) \Rightarrow Sin(\alpha-\beta)=2(\frac{12}{13})(\frac{5}{13})=\frac{120}{169}$$
$$Cos(\alpha+\beta)=Cos(\alpha)Cos(\beta)-Sin(\alpha)Sin(\beta) \Rightarrow Cos(\alpha+\beta)=-\frac{25}{169}-\frac{144}{169}=-1$$
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