Unit circle and trigonometric definitions

Parametric equations

The angle $\theta$ corresponds to the point on the unit circle

The tangent is $\tan \theta =\sin \theta /\cos \theta$ when $\cos \theta \ne 0$. Because $P$ lies on the circle of radius $1$, ${\sin }^2\theta +{\cos }^2\theta =1$ always holds.

• θ: the size of the angle; the button reads Angle display: degrees or Angle display: radians
• P: the corresponding point on the unit circle

Interaction

• Angle θ: drag the point on the circle or the slider to change $\theta$; $\cos \theta$, $\sin \theta$, and $\tan \theta$ update together
• Quadrant signs: on: mark each of the four quadrants with the sign of each function, the usual quadrant pattern
• Special angles: on: show the exact values at $30^{\circ }$, $45^{\circ }$, $60^{\circ }$, and the other marked angles; the marks stay fixed on the circle
• Perpendiculars from $P$ to the two axes show the lengths $\cos \theta$ and $\sin \theta$

What to notice

• $\sin \theta$ is the y-coordinate and $\cos \theta$ is the x-coordinate, and the sum of their squares is always $1$.
• $\tan \theta$ can be read as the length of a segment on the tangent to the unit circle. As $\theta \to 90^{\circ }{}^{-}$, the tangent tends to $+\infty$, and at $\theta =90^{\circ }$ it is undefined.
• Negative angles and angles past $360^{\circ }$ repeat a point on the circle, and that is where the periodicity comes from.

Related

• Sine law and cosine law
• Trigonometric identities and angle sums
• Geometric definitions and trigonometric identities

Further reading

• Unit circle (Wikipedia)
• Trigonometric functions (Wikipedia)