Trigonometric Values Formulas

Sine ($\sin$), cosine ($\cos$), and tangent ($\tan$) are the three primary trigonometric functions. They convert an angle into a ratio of triangle sides — and once you know any one ratio, the others follow.

Right-triangle definitions. For an acute angle $\theta$ in a right triangle: $\sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}$. The mnemonic SOH-CAH-TOA captures all three at once.

Unit-circle definitions. On the unit circle (radius 1, centered at the origin), the point at angle $\theta$ has coordinates $(\cos\theta,\sin\theta)$. So $\sin\theta$ is the y-coordinate, $\cos\theta$ is the x-coordinate, and $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$ is the slope through the origin at angle $\theta$. This is why sin, cos, tan extend to any real angle — positive, negative, or beyond 360°.

A full circle is 360° or $2\pi$ radians. To convert: $\text{radians} = \text{degrees}\times\dfrac{\pi}{180}$, and $\text{degrees} = \text{radians}\times\dfrac{180}{\pi}$.

Common values to memorize: $30°=\dfrac{\pi}{6}$, $45°=\dfrac{\pi}{4}$, $60°=\dfrac{\pi}{3}$, $90°=\dfrac{\pi}{2}$, $180°=\pi$, $270°=\dfrac{3\pi}{2}$, $360°=2\pi$.

For the five Q1 angles, $\sin$ follows a clean pattern: $\sin\theta=\dfrac{\sqrt{n}}{2}$ where $n=0,1,2,3,4$ for $\theta=0°,30°,45°,60°,90°$.

So $\sin 0°=\dfrac{\sqrt{0}}{2}=0$, $\sin 30°=\dfrac{\sqrt{1}}{2}=\dfrac{1}{2}$, $\sin 45°=\dfrac{\sqrt{2}}{2}$, $\sin 60°=\dfrac{\sqrt{3}}{2}$, $\sin 90°=\dfrac{\sqrt{4}}{2}=1$. For cosine, just read the same five values in reverse order.