# $\cos(x)=-\frac{24}{25}$ and $\tan(y) = \frac{9}{40}$. Calculate $\sin(x) \cos(y) + \cos(x) \sin(y)$ and $\cos(x) \cos(y) - \sin(x) \sin(y)$.

If $$\cos(x)= -\frac{24}{25}$$ and $$\tan(y) = \frac{9}{40}$$ for $$\frac{\pi}{2} < x < \pi$$ and $$\pi < y <\frac{3\pi}{2}$$. What is the value of $$\sin(x) \cos(y) + \cos(x) \sin(y)$$ and $$\cos(x) \cos(y) - \sin(x) \sin(y)$$?

Solution:

If $$\frac{\pi}{2} < x < \pi$$ then right triangle with acute angle $$x$$ facing west at second quadrant, the hypotenuse must be positive and the only negative is the adjacent. $$\cos(x)= -\frac{24}{25}$$ then $$adjacent=-24$$ and $$front=7$$ so $$\sin(x) = \frac{7}{25}$$.

If $$\pi < y <\frac{3\pi}{2}$$ then right triangle with acute angle $$x$$ facing west at third quadrant, the front and adjacent are negative. If $$\tan(y) = \frac{9}{40}$$ then the front is $$-9$$, the adjacent is $$-40$$, and the hypotenuse is $$41$$. So $$\sin(y) = -\frac{9}{41}$$ and $$\cos(y)=-\frac{40}{41}$$

The right triangle that we consider is the one with the adjacent side on the $$x$$-axis right..?

• Everything you are doing is correct. So all you have to do is plug in your values. – Zarrax Aug 22 '19 at 5:24

$$\hspace{1cm}$$ $$\cos x=-\frac{24}{25}=\frac{\overbrace{-24}^{adjacent}}{25}, x\in \left(\frac{\pi}{2},\pi\right); \quad \tan y=\frac{9}{40}=\frac{\overbrace{-9}^{front}}{\underbrace{-40}_{adjacent}},y\in \left(\pi,\frac{3\pi}{2}\right).$$ Also note: $$\cos x=-\cos(\pi -x)=-\frac{24}{25} \Rightarrow \cos (\pi -x)=\frac{24}{25}; \quad \sin (\pi -x)=\frac{7}{25}=\sin x.\\ \tan y=\tan(y-\pi)=\frac{9}{40}=\frac{-9}{-40} \Rightarrow \begin{cases}\sin y=-\sin(y-\pi)=-\frac{9}{41} \Rightarrow \sin (y-\pi)=\frac9{41}\\ \cos y=-\cos(y-\pi)=-\frac{40}{41} \Rightarrow \cos (y-\pi)=\frac{40}{41}\end{cases}$$
You can observe that $$\lvert\sin x\rvert=\sqrt{1-\cos^2x}=\sqrt{1-\frac{24^2}{25^2}}=\frac{7}{25}$$ Since $$\pi/2, we conclude that $$\sin x=7/25$$.
Also $$\lvert\cos y\rvert=\sqrt{\frac{1}{1+\tan^2y}}=\frac{40}{41}$$ and from $$\pi we conclude that $$\cos y=-40/41$$. Thus $$\sin y=\tan y\cos y=-9/41$$