Exercises: Sine, Cosine and Tangent on the unit circle

The cosine is positive in the first quadrant and in the fourth quadrant.

1st angle

$\cos \left(\gamma \right)=\frac{1}{2}\sqrt{2}$

${\gamma }_1={45}^{\circ }$

2nd angle

$\cos ({360}^{\circ }-{\gamma }_1)=\frac{1}{2}\sqrt{2}$

$\cos ({360}^{\circ }-{45}^{\circ })=\frac{1}{2}\sqrt{2}$

${\gamma }_2={315}^{\circ }$

3rd angle

$\cos ({360}^{\circ }+{\gamma }_1)=\frac{1}{2}\sqrt{2}$

$\cos ({360}^{\circ }+{45}^{\circ })=\frac{1}{2}\sqrt{2}$

${\gamma }_3={405}^{\circ }$

4th angle

$\cos ({720}^{\circ }-{\gamma }_1)=\frac{1}{2}\sqrt{2}$

$\cos ({720}^{\circ }-{45}^{\circ })=\frac{1}{2}\sqrt{2}$

${\gamma }_3={675}^{\circ }$

5th angle

$\cos (0^{\circ }-{\gamma }_1)=\frac{1}{2}\sqrt{2}$

$\cos (0^{\circ }-{45}^{\circ })=\frac{1}{2}\sqrt{2}$

${\gamma }_4=-{45}^{\circ }$

$\sin \left(\frac{x}{2}\right)=1$

The sine is positive in the first and second quadrant.

$y\in [-\mathrm{\pi };3\mathrm{\pi }]$

1st angle in radiant

$\frac{x}{2}=y$

$\sin \left(y\right)=1$

$y_1=\frac{1}{2}\mathrm{\pi }$

$x_1=\mathrm{\pi }$

2nd angle in radiant

$\sin (2\mathrm{\pi }+y_1)=1$

$\sin (2\mathrm{\pi }+\frac{\mathrm{\pi }}{2})=1$

${\gamma }_2=2.5\mathrm{\pi }$

$x=5\mathrm{\pi }$

3rd angle in radiant

$\sin (0-y_1)=1$

$\sin (-\frac{\mathrm{\pi }}{2})=1$

$y_3=-\frac{1}{2}\mathrm{\pi }$

$x=-\mathrm{\pi }$

$\sin \left(x\right)=-2$

Can never hold, since $-1\le \sin \left(x\right)\le 1$