Trigonometry calculator

What is a trigonometry calculator?

Trigonometry is the branch of mathematics that connects the angles of a triangle to the lengths of its sides. The trigonometry calculator takes a single angle and returns the six trigonometric functions defined for it: sine, cosine, tangent, cotangent, secant, and cosecant. You can enter the angle in degrees or radians, and the calculator handles the conversion for you.

These six functions describe ratios that stay constant for a given angle, no matter how large or small the triangle. That is what makes them so powerful: once you know the angle, the ratios are fixed, and they reappear everywhere from surveying and navigation to physics, signal processing, and computer graphics.

Key concepts

• Angle (θ) — the input angle, measured in degrees (a full turn is 360°) or radians (a full turn is $2\pi$).
• Sine and cosine — the two foundational functions; on the unit circle, $\cos\theta$ is the x-coordinate and $\sin\theta$ is the y-coordinate of the point at angle θ.
• Tangent — the ratio $\sin\theta / \cos\theta$, equal to the slope of the radius line at angle θ.
• Reciprocal functions — cotangent, secant, and cosecant are the reciprocals of tangent, cosine, and sine respectively.

How does the calculator work?

Imagine a circle of radius 1 centered at the origin — the unit circle. A point on the circle at angle θ (measured counterclockwise from the positive x-axis) has coordinates $(\cos\theta, \sin\theta)$. Every trigonometric function is built from these two coordinates.

Formulas

The two primary functions are the coordinates of the point on the unit circle:

The remaining four functions are ratios and reciprocals of these:

Because division by zero is undefined, some functions have no value at certain angles. Tangent and secant are undefined wherever $\cos\theta = 0$ (for example 90° and 270°), while cotangent and cosecant are undefined wherever $\sin\theta = 0$ (for example 0° and 180°). The calculator leaves those outputs blank.

Worked examples

Example 1: angle of 30°

At 30° the values are exact and well known:

Example 2: angle of 45°

At 45° sine and cosine are equal, so tangent and cotangent are both 1:

Example 3: angle of 90°

At 90° the point sits at the top of the unit circle, so $\cos 90° = 0$:

Here $\tan 90°$ and $\sec 90°$ are undefined because they divide by $\cos 90° = 0$.

Example 4: angle of 1 radian

Switching the input unit to radians and entering 1:

Practical uses

• Surveying and navigation — turning a measured angle of elevation into a height or a horizontal distance.
• Physics and engineering — resolving forces, velocities, or alternating-current signals into perpendicular components.
• Computer graphics and games — rotating points, aiming projectiles, and animating circular motion all rely on sine and cosine.
• Astronomy — relating the apparent angle of a star to distances using the same ratios.
• Triangle solving — once you have a function value, the right triangle angle calculator helps recover the angles of an actual triangle.

Notes

• Make sure the input unit matches your angle: entering 90 in radian mode is a completely different angle from 90 degrees.
• To convert an angle between degrees, radians, and gradians on its own, use the angle unit converter.
• Blank outputs mean the function is undefined at that angle (a division by zero), not that the calculator failed.
• The six functions repeat periodically, so an angle and that angle plus a full turn give identical results.
• Once you know an angle, you can feed it into the sector area calculator to size a circular wedge.