fagiolinow8xk

2023-02-22

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

Juliette Mahoney

Beginner2023-02-23Added 6 answers

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

$\mathrm{cos}\alpha =2\mathrm{sin}\alpha$

${\mathrm{sin}}^{2}\alpha +{\mathrm{cos}}^{2}\alpha =1$

$\mathrm{sin}}^{2}\alpha +4{\mathrm{sin}}^{2}\alpha =1\Rightarrow {\mathrm{sin}}^{2}\alpha =\frac{1}{5}\Rightarrow \mathrm{sin}\alpha =\frac{\sqrt{5}}{5$

$\mathrm{cos}\alpha =2\mathrm{sin}\alpha =2\frac{\sqrt{5}}{5}$

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

$\mathrm{cos}\alpha =2\mathrm{sin}\alpha$

${\mathrm{sin}}^{2}\alpha +{\mathrm{cos}}^{2}\alpha =1$

$\mathrm{sin}}^{2}\alpha +4{\mathrm{sin}}^{2}\alpha =1\Rightarrow {\mathrm{sin}}^{2}\alpha =\frac{1}{5}\Rightarrow \mathrm{sin}\alpha =\frac{\sqrt{5}}{5$

$\mathrm{cos}\alpha =2\mathrm{sin}\alpha =2\frac{\sqrt{5}}{5}$