Reciprocal Trig Ratios

A reciprocal is 1 divided by a non-zero number. For example, the reciprocal of $2/3$ is $3/2$. Cosecant, secant and cotangent use the familiar triangle ratios with numerator and denominator exchanged.

$\theta$ (theta) labels the angle. The hypotenuse is opposite the right angle; the opposite and adjacent sides are named relative to $\theta$.

The domain is the set of allowed inputs. Division by zero is undefined. Cosecant and cotangent need $\sin\theta\ne0$; secant needs $\cos\theta\ne0$.

You may also use $\cot\theta=1/\tan\theta$ when tangent is defined and non-zero. At $90^\circ$, use $\cot90^\circ=0/1=0$, even though $\tan90^\circ$ is undefined.

A reciprocal changes a value, not an angle. On a calculator, $\sin^{-1}$ finds an angle from a sine value; it does not mean cosecant. To evaluate cosecant, enter $1\div\sin(\theta)$.

The angle $\theta$ can extend beyond a right triangle. On the unit circle, measured anticlockwise from the positive horizontal axis, cosine is the horizontal coordinate and sine is the vertical coordinate. Negative angles turn clockwise.

Radians measure an angle as arc length divided by radius. A full turn is $360^\circ=2\pi$ radians; $180^\circ=\pi$ radians.

The four quadrants are the four regions between the axes. In order, the signs of $(\cos\theta,\sin\theta)$ are $(+,+),(-,+),(-,-),(+,-)$. A non-zero number and its reciprocal have the same sign.

For an exact angle, use exact values: $\sin30^\circ=1/2$, $\cos30^\circ=\sqrt3/2$, and $\sin45^\circ=\cos45^\circ=\sqrt2/2$. At $60^\circ$, the sine and cosine values swap. For other angles, use the calculator in the question's degree or radian mode.