Free Sine Cosine Tangent Calculator

The Sin Cos Tan Calculator, also known as a trigonometry calculator, is a free online tool that lets you compute the values of sine, cosine, and tangent for any angle in degrees or radians. Beyond simply returning a numeric result, this trigonometric functions calculator helps you understand how these three fundamental functions relate to right triangles and the unit circle.

The Core Trigonometric Functions

Sine (sin), cosine (cos), and tangent (tan) are the building blocks of trigonometry. They map an angle to a ratio of sides in a right triangle or to coordinates on a unit circle. On a unit circle (radius = 1), the sine of an angle is the vertical projection of the corresponding radius, while the cosine is the horizontal projection. The tangent, at the same angle, is the length of the segment drawn from the point on the circle to the horizontal axis, tangent to the circle.

Key Properties

• Periodicity: Sine and cosine have a period of $360^{\circ}$ (or $2\pi$ radians). Tangent, however, completes a full cycle every $180^{\circ}$.
• Range: Sine and cosine always fall between $-1$ and $1$. Tangent, on the other hand, can take any real value, from $-\infty$ to $+\infty$.
• Phase shift: Sine and cosine are out of phase by $90^{\circ}$; $\sin(\theta) = \cos(90^{\circ} - \theta)$.
• Quadrant signs: The signs of sine and cosine change depending on which quadrant the angle lies in, following the familiar CAST rule.

These properties make the trigonometric functions periodic, bounded (for sin/cos), and useful for modeling oscillatory phenomena.

Right Triangle Trigonometry

In any right triangle, the three trigonometric functions for an acute angle can be expressed as simple side ratios:

$$\sin(\alpha) = \dfrac{\text{opposite}}{\text{hypotenuse}},\qquad \cos(\alpha) = \dfrac{\text{adjacent}}{\text{hypotenuse}},\qquad \tan(\alpha) = \dfrac{\text{opposite}}{\text{adjacent}}$$

Here, "opposite" is the side farthest from the angle, and "adjacent" is the side next to the angle (excluding the hypotenuse). This “SOH CAH TOA” mnemonic helps you recall the formulas quickly.

Example: 3-4-5 Triangle

A classic right triangle with sides 3, 4, and 5. For the angle opposite the side of length 3:

• $\sin(\alpha) = \dfrac{3}{5} = 0.6$
• $\cos(\alpha) = \dfrac{4}{5} = 0.8$
• $\tan(\alpha) = \dfrac{3}{4} = 0.75$

The same ratios for the other acute angle can be found by swapping “opposite” and “adjacent.”

Using the Trig Values Calculator

The sine cosine tangent calculator on this page accepts any angle—in degrees or radians—and immediately displays the corresponding sin, cos, and tan values. It also computes relevant triangle sides if you provide at least two pieces of information (e.g., a side and an angle, or two sides). This makes it a handy tool for both students checking homework and professionals needing quick trig values.

You can also use the calculator to explore relationships like the Pythagorean identity:

$$\sin^{2}(\alpha) + \cos^{2}(\alpha) = 1$$

Or the quotient identity:

$$\tan(\alpha) = \dfrac{\sin(\alpha)}{\cos(\alpha)}$$

Keep in mind that tangent is undefined wherever $\cos(\alpha) = 0$, which happens at $90^{\circ} + k\cdot 180^{\circ}$ (or $\pi/2 + k\pi$ radians). The calculator will indicate these singularities clearly.

With these tools, you can quickly calculate sine cosine tangent for any angle and deepen your understanding of right triangle trigonometry.