Free Trig Calculator

Overview of the Trigonometry Calculator

The Trigonometry Calculator is a multi‑function online tool that acts as a Trigonometric Functions Calculator, a Sine Cosine Tangent Calculator, and a Right Triangle Trigonometry solver — all in one interface. You can evaluate any of the six trigonometric functions at a given angle, or enter two known measurements from a right triangle to find its missing sides and angles. This makes the tool ideal for students, teachers, and professionals who need accurate trig function values without manual calculations.

Understanding Sine and Cosine

Trigonometric functions are defined using a right triangle placed inside a unit circle (radius = 1). For an angle $\theta$ measured counter‑clockwise from the positive $x$-axis:

• $\sin\theta$ corresponds to the vertical coordinate of the point where the terminal side meets the circle.
• $\cos\theta$ corresponds to the horizontal coordinate of that point.

Because the circumference radius is exactly one, both $\sin\theta$ and $\cos\theta$ are always between $-1$ and $1$. As the angle rotates through the four quadrants, the signs of these coordinates change:

Quadrant	Angle Range	$\sin\theta$	$\cos\theta$
I	$0^\circ$–$90^\circ$	$+$	$+$
II	$90^\circ$–$180^\circ$	$+$	$-$
III	$180^\circ$–$270^\circ$	$-$	$-$
IV	$270^\circ$–$360^\circ$	$-$	$+$

For example, at $135^\circ$ (second quadrant) the horizontal coordinate is negative, so $\cos 135^\circ = -\frac{\sqrt{2}}{2}$, while $\sin 135^\circ = +\frac{\sqrt{2}}{2}$.

Periodicity and Negative Angles

Trigonometric functions repeat their values every full rotation ($360^\circ$ or $2\pi$ rad). This property is captured by the periodic identities for any integer $n$:

$$\sin(\theta + 2\pi n) = \sin\theta,\qquad \cos(\theta + 2\pi n) = \cos\theta$$

Thus, an angle of $450^\circ$ ($= 90^\circ + 360^\circ$) yields the same sine and cosine as $90^\circ$. The Trig Calculator automatically reduces any angle to its equivalent within the standard range, so you can input values beyond $360^\circ$ and still obtain correct results.

Negative angles simply denote clockwise rotation. For instance, $-90^\circ$ lands at the same position as $270^\circ$. Hence $\sin(-90^\circ) = -1$ and $\cos(-90^\circ) = 0$.

The Four Derived Functions: Tan, Cot, Sec, Csc

Once sine and cosine are known, the remaining trig function values come from straightforward identities:

• Tangent: $\displaystyle \tan\theta = \frac{\sin\theta}{\cos\theta}$
• Cosecant: $\displaystyle \csc\theta = \frac{1}{\sin\theta}$
• Secant: $\displaystyle \sec\theta = \frac{1}{\cos\theta}$
• Cotangent: $\displaystyle \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$

These definitions are valid wherever the denominator is non‑zero. For example, $\tan 90^\circ$ is undefined because $\cos 90^\circ = 0$; the calculator indicates such cases with “undefined” or “infinity”.

Common Trigonometric Values

The following table lists sine, cosine, and tangent for frequently used angles. These values can be derived from the unit circle and are built into the calculator for quick reference.

Angle	$\sin\theta$	$\cos\theta$	$\tan\theta$
$0^\circ$	$0$	$1$	$0$
$30^\circ$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
$45^\circ$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
$60^\circ$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
$90^\circ$	$1$	$0$	undefined
$180^\circ$	$0$	$-1$	$0$
$270^\circ$	$-1$	$0$	undefined
$360^\circ$	$0$	$1$	$0$

Right Triangle Trigonometry in Practice

The unit‑circle definitions extend to any right triangle. For an acute angle $\alpha$, the sides are labeled as opposite, adjacent, and hypotenuse. The primary ratios (SOH CAH TOA) are:

$$\sin\alpha = \frac{\text{opposite}}{\text{hypotenuse}},\quad \cos\alpha = \frac{\text{adjacent}}{\text{hypotenuse}},\quad \tan\alpha = \frac{\text{opposite}}{\text{adjacent}}$$

Their reciprocal counterparts follow directly:

$$\csc\alpha = \frac{\text{hypotenuse}}{\text{opposite}},\quad \sec\alpha = \frac{\text{hypotenuse}}{\text{adjacent}},\quad \cot\alpha = \frac{\text{adjacent}}{\text{opposite}}$$

If the right triangle contains another acute angle $\beta$, the leg labels swap, but the formulas remain the same.

Putting the Calculator to Work

The tool accepts angles in degrees or radians and instantly displays all six trigonometric function values — no need for separate sine cosine tangent calculators. For triangle problems, you provide two known values (one side + one angle, or two sides), and the right triangle trigonometry solver computes the remaining sides and the other acute angle. The results are presented together with the formulas used, making the process transparent and educational.

All calculations rely on the fundamental trigonometric identities — periodic, reciprocal, and quotient — ensuring reliable outputs for any input, whether it consists of large angles, negative rotations, or everyday triangle measurements.