Sine and cosine on the unit circle

Every number $\displaystyle t$ has a point $\displaystyle P(t)$ on the unit circle. Its two coordinates are two new functions of $\displaystyle t$:

• the cosine of $\displaystyle t$, $\displaystyle \cos t$ – the $\displaystyle x$-coordinate of $\displaystyle P(t)$;
• the sine of $\displaystyle t$, $\displaystyle \sin t$ – the $\displaystyle y$-coordinate of $\displaystyle P(t)$.

On the board: the point $\displaystyle P(2)$, and the dashed lines from it to the axes. $\displaystyle P(2)$ is in the second quadrant, so:

• $\displaystyle \cos2$ is negative – about $\displaystyle -0.42$;
• $\displaystyle \sin2$ is positive – about $\displaystyle 0.91$.

A calculator set to radians (RAD) gives these values.

The four points where the circle meets the axes give the values at once:

The sign of a coordinate depends on the quadrant the point is in:

• first quadrant: $\displaystyle \cos t>0$, $\displaystyle \sin t>0$;
• second quadrant: $\displaystyle \cos t<0$, $\displaystyle \sin t>0$;
• third quadrant: $\displaystyle \cos t<0$, $\displaystyle \sin t<0$;
• fourth quadrant: $\displaystyle \cos t>0$, $\displaystyle \sin t<0$.

Every point on the unit circle is at most 1 away from each axis, so:

The point $\displaystyle (\cos t,\sin t)$ is on the circle $\displaystyle {x^2+y^2}={1}$. Substitute it in the circle’s equation:

($\displaystyle \sin^2t$ means $\displaystyle (\sin t)^2$.) This is an identity: it holds for every number $\displaystyle t$. An equation, by contrast, holds only for some numbers – the ones you look for when you solve it.

The example below finds $\displaystyle \cos t$, where $\displaystyle {\sin t}={\frac{3}{5}}$ and $\displaystyle P(t)$ is in the second quadrant.

• $\displaystyle {\cos^2t}={1-\frac{9}{25}}={\frac{16}{25}}$, so $\displaystyle {\cos t}={\frac{4}{5}}$ or $\displaystyle {\cos t}={-\frac{4}{5}}$.
• In the second quadrant the cosine is negative: $\displaystyle {\cos t}={-\frac{4}{5}}$.

Work out without a calculator.

Decide without a calculator the sign of each number, and explain by the quadrant:
$\displaystyle \sin2$, $\displaystyle \cos2$, $\displaystyle \sin4$, $\displaystyle \cos5$, $\displaystyle \sin(-1)$.

$\displaystyle {\sin t}={-\frac{5}{13}}$, and the point $\displaystyle P(t)$ is in the third quadrant.

$\displaystyle x$ is $\displaystyle \cos t$. Solve the equation the basic identity gives.

Which of the solutions is $\displaystyle \cos t$? Explain.

Is there a number $\displaystyle t$ that makes both equalities true?

$\displaystyle {\cos t}={0.6}$ and $\displaystyle {\sin t}={0.8}$. Work out $\displaystyle \cos^2t+\sin^2t$.

$\displaystyle {\cos t}={0.5}$ and $\displaystyle {\sin t}={0.5}$. Work out $\displaystyle \cos^2t+\sin^2t$.

What is the answer in each part, and why? And what about $\displaystyle {\sin t}={1.2}$?