Unit circle explorer

Drag the point around the circle, or use the slider, to set the angle $\theta$. The horizontal leg of the triangle is $\cos\theta$ and the vertical leg is $\sin\theta$. The two graphs show the same numbers as heights, so the point traces one full wave of each as it goes around once.

Things to try

• Turn on snapping and step through the special angles with the arrow keys. The same few sizes, $\frac{1}{2}$, $\frac{\sqrt{2}}{2}$ and $\frac{\sqrt{3}}{2}$, come back in every quadrant with different signs.
• Compare $30^\circ$, $150^\circ$, $210^\circ$ and $330^\circ$. All four have the reference angle $30^\circ$, so their sines and cosines have the same size and differ only in sign.
• Move to $90^\circ$ and then to $270^\circ$. The point sits on the $y$ axis, the cosine is $0$ and the tangent is undefined. One degree to either side the tangent is large, with opposite signs on the two sides.
• Watch the cosine graph while the point moves through quadrant II. The horizontal leg points left, so the cosine is negative even though the point is still in the upper half of the circle.
• Switch to radians and find $\frac{\pi}{2}$, $\pi$ and $\frac{3\pi}{2}$ on the graphs. The sine wave crosses zero at $0$, $\pi$ and $2\pi$, exactly where the point is on the $x$ axis.
• Pick an angle away from the special ones, square the readouts for $\cos\theta$ and $\sin\theta$ and add them. The total is $1$, up to rounding.

What you are seeing

Sine, cosine and tangent

Put the angle $\theta$ in standard position, with its vertex at the origin and its initial side along the positive $x$ axis. Its terminal side meets the unit circle $x^2 + y^2 = 1$ at exactly one point, and the coordinates of that point are the cosine and the sine of $\theta$.

The radius from the origin to $P$ has length $1$. A vertical line from $P$ down or up to the $x$ axis makes a right triangle with hypotenuse $1$, horizontal leg $\cos\theta$ and vertical leg $\sin\theta$. The legs are signed, so a leg that points left or down counts as negative.

Since $P$ is on the circle, its coordinates satisfy the Pythagorean identity at every angle.

The tangent is the slope of the radius.

It is undefined at $90^\circ$ and $270^\circ$, where the radius is vertical and the cosine is $0$.

Reference angles and signs

The reference angle is the acute angle between the terminal side and the $x$ axis. It fixes the size of each value, and the quadrant fixes the sign. The dashed arc in the picture marks it.

For example, the angle $150^\circ$ is in quadrant II with reference angle $30^\circ$. Cosine is negative there and sine is positive.

An angle whose terminal side lies on an axis, such as $90^\circ$ or $180^\circ$, is quadrantal. It is in no quadrant and has no reference angle, so the explorer shows none. Its values come straight from the points $(1, 0)$, $(0, 1)$, $(-1, 0)$ and $(0, -1)$.

Exact values

The special angles are the multiples of $30^\circ$ and $45^\circ$. Away from the axes their reference angles are $30^\circ$, $45^\circ$ and $60^\circ$, and the sizes come from the two special right triangles.

At the special angles the explorer gives these exact values with the right sign, along with their decimals. At every other angle it shows decimals rounded to four places.

Radians and coterminal angles

An angle of $\theta$ radians cuts off an arc of length $\theta$ on the unit circle. A full turn of $360^\circ$ goes once around the circle, an arc of length $2\pi$, so half a turn gives

The explorer writes radians as a multiple of $\pi$ when the fraction has a denominator of $12$ or less, such as $\frac{5\pi}{6}$ for $150^\circ$ or $\frac{\pi}{9}$ for $20^\circ$. Other angles appear as rounded decimals with the exact multiple of $\pi$ alongside, so $37^\circ$ shows as $0.646$ next to its exact value $\frac{37\pi}{180}$.

Angles that differ by whole turns, such as $30^\circ$, $390^\circ$ and $-330^\circ$, are coterminal. They end at the same point of the circle, so they have the same sine, cosine and tangent.

That is why the slider only needs one turn, from $0^\circ$ up to $359^\circ$.

The graphs

As $\theta$ runs from $0$ to $2\pi$, the height of $P$ traces one period of $y = \sin\theta$ and its horizontal position traces one period of $y = \cos\theta$. The colored bar on each graph is the matching leg of the triangle, stood on end. After $2\pi$ the point starts the same trip again, which is why both graphs repeat with period $2\pi$.

These ideas are built up in the lessons Special Right Triangles, General Angles and Reference Angles, The Unit Circle and Graphs of Sine and Cosine. Radian measure and coterminal angles come from Angles, Radians, and Coterminal Angles. The formulas are collected on the precalculus reference sheet, and every lesson is listed on the Precalculus course page. For test preparation, the SAT Math lessons Degrees, Radians and the Unit Circle and Sine, Cosine and Tangent in Right Triangles use the same ideas.