Free Trigonometric Functions Calculator

The Trigonometric Functions Calculator offers a complete solution as a six trig functions calculator, returning the values of sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot) for any entered angle. Whether you require a quick sin cos tan calculator or a dedicated csc sec cot calculator, this online trigonometry calculator delivers accurate angle trigonometric values in both degrees and radians. Below you will find the definitions of these trigonometric ratios, the helpful SOHCAHTOA mnemonic, and a guide to using this tool effectively.

Trigonometric Ratios in a Right Triangle

For an acute angle $\alpha$ in a right triangle, the three primary functions are based on side lengths:

• Hypotenuse – the side opposite the right angle and the longest side.
• Adjacent side – the leg that together with the hypotenuse forms $\alpha$.
• Opposite side – the leg directly facing $\alpha$.

The fundamental ratios are:

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

Reciprocal Functions

Each basic function has a reciprocal with its own name and use:

• $\csc(\alpha) = \dfrac{1}{\sin(\alpha)} = \dfrac{\text{hypotenuse}}{\text{opposite}}$
• $\sec(\alpha) = \dfrac{1}{\cos(\alpha)} = \dfrac{\text{hypotenuse}}{\text{adjacent}}$
• $\cot(\alpha) = \dfrac{1}{\tan(\alpha)} = \dfrac{\text{adjacent}}{\text{opposite}}$

Because they are reciprocals, cosecant and secant satisfy $|\csc(\alpha)|\ge 1$ and $|\sec(\alpha)|\ge 1$, while cotangent can be any real number. Together these six functions form the standard set of trigonometric ratios.

Remembering with SOHCAHTOA

The mnemonic SOHCAHTOA helps recall the three basic definitions:

• SOH – Sine equals Opposite over Hypotenuse.
• CAH – Cosine equals Adjacent over Hypotenuse.
• TOA – Tangent equals Opposite over Adjacent.

Many learners create personal phrases (e.g., “Some Old Horses Can Always Hear Their Owners Approach”) to fix these relationships in memory. Regardless of the phrase, the underlying formulas remain unchanged.

Visual Overview of the Functions

The calculator provides interactive plots of all six functions over $[-2\pi, 2\pi]$, showing the familiar sine and cosine waves, the steep tangent curve, and the reciprocal shapes. These graphs help illustrate periodicity, asymptotes, and other properties.

Step‑by‑Step Usage Example

To obtain the six values for a given angle:

1. Choose the unit: degrees, radians, or π‑radians.
2. Enter the angle. For instance, to compute for $\pi/3$ rad, select “π rad” and input $1/3$.
3. Press calculate.

The tool instantly shows:

• $\sin(\pi/3) = \dfrac{\sqrt{3}}{2} \approx 0.8660$
• $\cos(\pi/3) = \dfrac{1}{2} = 0.5$
• $\tan(\pi/3) = \sqrt{3} \approx 1.7321$
• $\csc(\pi/3) = \dfrac{2}{\sqrt{3}} \approx 1.1547$
• $\sec(\pi/3) = 2$
• $\cot(\pi/3) = \dfrac{1}{\sqrt{3}} \approx 0.5774$

If the angle is acute, a right‑triangle diagram accompanies the results for extra clarity.

Why This Tool?

Instead of manually computing each ratio or switching between separate calculators, this single six trig functions calculator provides every angle trigonometric value in one place. Its support for degrees and radians, along with immediate output, makes it ideal for students, educators, and professionals working with trigonometry or related fields.