Free Trig Degree Calculator

Trig Degree Calculator – Free Online Trigonometric Function Tool

The Trig Degree Calculator is a free online trigonometric functions calculator that instantly evaluates all six basic trig functions—$\sin$, $\cos$, $\tan$, $\cot$, $\sec$, $\csc$—for any angle you input. Whether you work in degrees or radians, this sin cos tan calculator returns both exact expressions (using fractions and radicals) and decimal approximations, making it a versatile angle calculator trig for homework, engineering, or quick verification. As a six trig functions calculator, it eliminates the need for manual lookup tables.

How to Use the Calculator

Using the tool is straightforward: enter an angle in degrees (e.g., 45°) or radians (e.g., $\pi/4$) and select the appropriate mode. The calculator immediately displays all six trigonometric values. You can switch between degree and radian input at any time.

Definition of the Trigonometric Functions

For a given angle $\theta$ in a right triangle, the primary functions are defined as:

• $\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$
• $\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$
• $\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$

Their reciprocals extend the set:

• $\csc\theta = \dfrac{1}{\sin\theta}$
• $\sec\theta = \dfrac{1}{\cos\theta}$
• $\cot\theta = \dfrac{1}{\tan\theta} = \dfrac{\cos\theta}{\sin\theta}$

Core Identities for Deriving Values

From the sine and cosine alone, the other four functions can be obtained through the following relationships:

$$\tan\theta = \frac{\sin\theta}{\cos\theta},\qquad \cot\theta = \frac{\cos\theta}{\sin\theta},\qquad \sec\theta = \frac{1}{\cos\theta},\qquad \csc\theta = \frac{1}{\sin\theta}.$$

For example, for $\theta = 30^\circ$:

$$\sin30^\circ = \frac{1}{2},\quad \cos30^\circ = \frac{\sqrt{3}}{2},\quad \tan30^\circ = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}},\quad \cot30^\circ = \frac{\sqrt{3}/2}{1/2} = \sqrt{3},\quad \sec30^\circ = \frac{1}{\sqrt{3}/2} = \frac{2}{\sqrt{3}},\quad \csc30^\circ = \frac{1}{1/2} = 2.$$

Reference Table for Standard Angles

The table below gives the sine and cosine values for the most frequently encountered angles. The remaining functions can be derived using the identities above.

Angle (θ)	sin(θ)	cos(θ)
0°	0	1
30°	$\dfrac{1}{2}$	$\dfrac{\sqrt{3}}{2}$
45°	$\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{2}}{2}$
60°	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{2}$
90°	1	0
120°	$\dfrac{\sqrt{3}}{2}$	$-\dfrac{1}{2}$
135°	$\dfrac{\sqrt{2}}{2}$	$-\dfrac{\sqrt{2}}{2}$
150°	$\dfrac{1}{2}$	$-\dfrac{\sqrt{3}}{2}$
180°	0	–1
210°	$-\dfrac{1}{2}$	$-\dfrac{\sqrt{3}}{2}$
225°	$-\dfrac{\sqrt{2}}{2}$	$-\dfrac{\sqrt{2}}{2}$
240°	$-\dfrac{\sqrt{3}}{2}$	$-\dfrac{1}{2}$
270°	–1	0
300°	$-\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{2}$
315°	$-\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{2}}{2}$
330°	$-\dfrac{1}{2}$	$\dfrac{\sqrt{3}}{2}$
360°	0	1

The exact expressions (fractions and radicals) preserve precision, while the calculator also displays decimal approximations for quick reference.

Undefined Values

Some function‑angle combinations require division by zero, which has no mathematical meaning. These cases are marked und (undefined) in the results. For instance:

• $\tan90^\circ = \dfrac{1}{0}$ → undefined
• $\cot0^\circ = \dfrac{1}{0}$ → undefined
• $\sec90^\circ = \dfrac{1}{0}$ → undefined
• $\csc0^\circ = \dfrac{1}{0}$ → undefined

The same holds for angles where the denominator in the defining identity becomes zero.

Sign of Trigonometric Functions by Quadrant

When working on the coordinate plane, knowing the sign of each function in the four quadrants is essential. The table below summarizes the sign pattern.

Quadrant	sin(θ)	cos(θ)	tan(θ)	cot(θ)	sec(θ)	csc(θ)
I	+	+	+	+	+	+
II	+	–	–	–	–	+
III	–	–	+	+	–	–
IV	–	+	–	–	+	–

A useful mnemonic is “All Students Take Calculus”: All six are positive in Quadrant I; Sine (and cosecant) positive in Quadrant II; Tangent (and cotangent) positive in Quadrant III; Cosine (and secant) positive in Quadrant IV.

Practical Notes

• The calculator provides both the exact symbolic result (such as $\frac{\sqrt{3}}{2}$) and its decimal equivalent (≈0.8660), giving you flexibility for symbolic manipulation or quick numerical work.
• If you need to compute values for a non‑standard angle, simply enter it in the tool; the on‑the‑fly computation avoids interpolation or manual chart reading.