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The Trigonometry Calculator computes all six trigonometric functions for any given angle. Enter an angle in degrees or radians, and instantly obtain the values of sine, cosine, tangent, cosecant, secant, and cotangent.
Trigonometry is the branch of mathematics that studies the relationships between the sides and angles of triangles. The six trigonometric functions are defined using a right triangle or, more generally, using the unit circle. For a right triangle with angle $$\theta$$, opposite side $$a$$, adjacent side $$b$$, and hypotenuse $$c$$:
$$\sin\theta = \frac{a}{c}, \quad \cos\theta = \frac{b}{c}, \quad \tan\theta = \frac{a}{b}$$
The reciprocal functions are derived directly from these primary functions:
$$\csc\theta = \frac{1}{\sin\theta} = \frac{c}{a}, \quad \sec\theta = \frac{1}{\cos\theta} = \frac{c}{b}, \quad \cot\theta = \frac{1}{\tan\theta} = \frac{b}{a}$$
Using the unit circle (a circle of radius 1 centered at the origin), trigonometric functions extend to all real numbers. The point on the unit circle at angle $$\theta$$ from the positive x-axis has coordinates $$(\cos\theta, \sin\theta)$$, which elegantly defines sine and cosine for any angle, including negative angles and angles beyond 360 degrees.
Angles can be measured in degrees or radians. A full rotation is $$360°$$ or $$2\pi$$ radians. The conversion formula is:
$$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$
This calculator handles both units automatically. Key special angles include $$0°, 30°, 45°, 60°,$$ and $$90°$$, whose trigonometric values are frequently used in geometry, physics, and engineering. For example, $$\sin 30° = 0.5$$, $$\cos 60° = 0.5$$, and $$\tan 45° = 1$$.
Trigonometric functions are periodic: sine and cosine repeat every $$2\pi$$ radians (360°), while tangent and cotangent repeat every $$\pi$$ radians (180°). Some functions are undefined at certain angles — tangent is undefined at $$90°$$ and $$270°$$ (where cosine equals zero), and cosecant is undefined at $$0°$$ and $$180°$$ (where sine equals zero).
Applications of trigonometry span virtually every scientific and engineering discipline: calculating distances and heights in surveying, analyzing alternating current in electrical engineering, modeling wave phenomena in physics, computing orbital mechanics in astronomy, and rendering 3D graphics in computer science. This calculator provides a quick reference for all six function values at once.
The calculator converts the input angle to radians (if given in degrees) using $$\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}$$. It then evaluates all six trigonometric functions. Reciprocal functions that would involve division by zero return undefined (NaN).
Each output represents a ratio. Sine and cosine always fall between $$-1$$ and $$1$$. Tangent and cotangent can take any real value. Cosecant and secant have magnitudes $$\geq 1$$. An undefined result means the function does not exist at that angle (e.g., $$\tan 90°$$).
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At 45°, sine equals cosine (both √2/2 ≈ 0.7071), and tangent equals cotangent (both 1). The reciprocal functions csc and sec both equal √2 ≈ 1.4142.
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π/6 radians equals 30°. sin(30°) = 0.5, cos(30°) = √3/2, tan(30°) = 1/√3. Cosecant is 2, secant is 2/√3, and cotangent is √3.
Degrees divide a full circle into 360 equal parts, while radians measure angles based on the radius of a circle. One full rotation equals $$360°$$ or $$2\pi$$ radians. To convert, multiply degrees by $$\frac{\pi}{180}$$. Radians are the standard unit in calculus and physics.
A function is undefined when its computation involves division by zero. Tangent is $$\frac{\sin\theta}{\cos\theta}$$, so it is undefined when $$\cos\theta = 0$$ (at 90° and 270°). Similarly, cosecant is undefined when $$\sin\theta = 0$$ (at 0° and 180°).
The unit circle is a circle of radius 1 centered at the origin of the coordinate plane. For any angle $$\theta$$, the point on the unit circle has coordinates $$(\cos\theta, \sin\theta)$$. This definition extends trigonometric functions to all real numbers, not just acute angles in right triangles.
The fundamental identity is $$\sin^2\theta + \cos^2\theta = 1$$. Dividing by $$\cos^2\theta$$ gives $$\tan^2\theta + 1 = \sec^2\theta$$. Dividing by $$\sin^2\theta$$ gives $$1 + \cot^2\theta = \csc^2\theta$$. These identities hold for all valid angles.
The reference angle is the acute angle formed with the x-axis. For angles in Quadrant II, subtract from 180°. For Quadrant III, subtract 180°. For Quadrant IV, subtract from 360°. The reference angle always lies between 0° and 90°, and trig function magnitudes equal those of the reference angle.
Use the mnemonic All Students Take Calculus: In Quadrant I, all functions are positive. In Quadrant II, only sine and cosecant. In Quadrant III, only tangent and cotangent. In Quadrant IV, only cosine and secant. This pattern follows from the signs of x and y coordinates.
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