0.5
1.04719755
rad
60
°
0.166667
0.5
1.04719755
rad
60
°
0.166667
The Cosine Calculator evaluates the cosine of any angle given in degrees or radians. Cosine is one of the three primary trigonometric functions and is essential in fields ranging from geometry and navigation to signal processing and quantum mechanics.
In a right triangle, cosine is defined as the ratio of the adjacent side to the hypotenuse:
$$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$$
On the unit circle, cosine corresponds to the x-coordinate of the point at angle $$\theta$$. Like sine, cosine is bounded between $$-1$$ and $$1$$. It reaches its maximum of 1 at $$\theta = 0°$$ and its minimum of $$-1$$ at $$\theta = 180°$$.
Cosine is an even function, meaning $$\cos(-\theta) = \cos\theta$$. This symmetry about the y-axis distinguishes it from sine, which is odd. Cosine has the same period of $$2\pi$$ radians (360°) as sine.
The standard exact values are:
$$\cos 0° = 1, \quad \cos 30° = \frac{\sqrt{3}}{2}, \quad \cos 45° = \frac{\sqrt{2}}{2}, \quad \cos 60° = \frac{1}{2}, \quad \cos 90° = 0$$
Cosine is intimately connected to sine through the co-function identity: $$\cos\theta = \sin(90° - \theta)$$. This relationship means that the cosine of an angle equals the sine of its complement. Together with the Pythagorean identity $$\sin^2\theta + \cos^2\theta = 1$$, these relationships form the backbone of trigonometric analysis.
The law of cosines generalizes the Pythagorean theorem to any triangle: $$c^2 = a^2 + b^2 - 2ab\cos C$$. When $$C = 90°$$, $$\cos 90° = 0$$, and the formula reduces to $$c^2 = a^2 + b^2$$, recovering the Pythagorean theorem. This law is fundamental in surveying, navigation, and computer graphics for calculating distances.
In physics, cosine appears in the definition of work: $$W = F \cdot d \cdot \cos\theta$$, where $$\theta$$ is the angle between the force and displacement vectors. It also governs the dot product of two vectors: $$\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta$$.
The cosine function also plays a key role in Fourier analysis, where any periodic signal can be decomposed into a sum of cosine (and sine) waves. The discrete cosine transform (DCT) is the foundation of JPEG image compression and MP3 audio encoding.
The Taylor series for cosine converges for all real numbers: $$\cos\theta = 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \frac{\theta^6}{6!} + \cdots$$ (with $$\theta$$ in radians). This series uses only even powers, reflecting the even symmetry of the function.
The calculator converts the input angle to radians if given in degrees, then evaluates $$\cos(\theta_{rad})$$. The inverse check applies $$\arccos$$ and converts the result to degrees, returning the principal value in $$[0°, 180°]$$.
The cosine value lies between $$-1$$ and $$1$$. Positive values occur in Quadrants I and IV (−90° to 90°), negative in Quadrants II and III. The inverse check returns the principal angle in $$[0°, 180°]$$, so negative input angles will map to their positive equivalent.
Inputs
Results
cos(60°) = 0.5 exactly. The angle in radians is π/3 ≈ 1.0472. The inverse check confirms 60°, which lies within arccos's principal range of [0°, 180°].
Inputs
Results
cos(300°) = cos(−60°) = cos(60°) = 0.5 because cosine is even. The inverse check returns 60°, not 300°, since arccos range is [0°, 180°].
$$\cos 0° = 1$$, $$\cos 90° = 0$$, $$\cos 180° = -1$$, $$\cos 270° = 0$$, $$\cos 360° = 1$$. Cosine starts at 1, drops to 0, reaches −1 at 180°, returns to 0 at 270°, and completes the cycle at 1.
A function is even if $$f(-x) = f(x)$$ for all $$x$$. Since $$\cos(-\theta) = \cos\theta$$, cosine satisfies this property. On the unit circle, reflecting an angle across the x-axis does not change the x-coordinate, which is cosine.
For a triangle with sides $$a, b, c$$ and angle $$C$$ opposite side $$c$$: $$c^2 = a^2 + b^2 - 2ab\cos C$$. This generalizes the Pythagorean theorem and works for any triangle, not just right triangles. It is used to find an unknown side when two sides and the included angle are known.
The derivative of $$\cos(x)$$ is $$-\sin(x)$$: $$\frac{d}{dx}\cos(x) = -\sin(x)$$. The negative sign means cosine decreases where sine is positive. The integral of cosine is $$\sin(x) + C$$.
The dot product of vectors $$\mathbf{a}$$ and $$\mathbf{b}$$ equals $$|\mathbf{a}||\mathbf{b}|\cos\theta$$, where $$\theta$$ is the angle between them. When $$\cos\theta = 0$$ (at 90°), the vectors are perpendicular. When $$\cos\theta = 1$$ (at 0°), they point in the same direction.
Because cosine is even, $$\cos(-\theta) = \cos\theta$$. For example, $$\cos(-45°) = \cos(45°) = \frac{\sqrt{2}}{2} \approx 0.7071$$. This symmetry property simplifies many calculations involving negative angles.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
