fagiolinow8xk

2023-02-22

What is sin and cos if tan = 1/2 and sin>0?

Juliette Mahoney

If the sine and tangent of an angle $\alpha$ are positive, the angle lies in the first quadrant, meaning both the sine and cosine are positive.
Since $\mathrm{tan}\alpha =\frac{\mathrm{sin}\alpha }{\mathrm{cos}\alpha }$
So
$\frac{\mathrm{sin}\alpha }{\mathrm{cos}\alpha }=\frac{1}{2}$
Trigonometry is based on
${\mathrm{sin}}^{2}\alpha +{\mathrm{cos}}^{2}\alpha =1$
So, let's use it for is
$\mathrm{cos}\alpha =2\mathrm{sin}\alpha$
${\mathrm{sin}}^{2}\alpha +{\mathrm{cos}}^{2}\alpha =1$
${\mathrm{sin}}^{2}\alpha +4{\mathrm{sin}}^{2}\alpha =1⇒{\mathrm{sin}}^{2}\alpha =\frac{1}{5}⇒\mathrm{sin}\alpha =\frac{\sqrt{5}}{5}$
$\mathrm{cos}\alpha =2\mathrm{sin}\alpha =2\frac{\sqrt{5}}{5}$

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