Exercises: Sine, Cosine and Tangent on the unit circle

Draw the coordinate system.

Construct the unit circle with your compass.

Use a triangle to draw a $60°$ angle with respect to the $x$-axis. Mark the intersection of the resulting straight line with the unit circle.

Draw the perpendicular to the $x$-axis that passes through the marked point.

Now you can measure the length of the sine with your ruler.

$\sin (60°)\approx 0.87$

In a right triangle with a $60°$ angle, the third interior angle has $30°$. In the unit circle the hypotenuse has length $1$.

Extend the triangle to an equilateral triangle by mirroring the given triangle at the edge with $\sin (60°)$. The triangle must be equilateral because it has three $60°$ angles.

In an equilateral triangle all sides have the same length, i.e. length $1$.

$\cos (60°)$ is exactly half the side length of the triangle, i.e. $\frac{1}{2}$.

Now use the Pythagorean theorem in the right triangle to calculate $\sin (60°)$.

Plug in the value for the cosine. ${(\cos (60°))}^2+{(\sin (60°))}^2=1^2$

${\left(\frac{1}{2}\right)}^2+{(\sin (60°))}^2=1$

$\frac{1}{4}+{(\sin (60°))}^2=1$ $|-\frac{1}{4}$

${(\sin (60°))}^2=\frac{3}{4}$ $|\sqrt{}$

$\sin (60°)=\pm \sqrt{\frac{3}{4}}$ since $\sin (60°)$ was applied above the $x$-axis, only the positive result comes into question.

$\sin (60°)=\sqrt{\frac{3}{4}}$

$\sin (60°)=\frac{1}{2}\sqrt{3}$