Trigonometric Calculator
Calculate Sine, Cosine, and Tangent for a given angle.
Degrees
Radians
Results for
Sine (sin)
Cosine (cos)
Tangent (tan)
Understanding Trigonometric Functions: Sine, Cosine, and Tangent
Trigonometry is a fundamental branch of mathematics that studies the relationships between the sides and angles of triangles. The core of trigonometry lies in its six functions: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Among these, sine, cosine, and tangent are the most commonly used. These functions are defined using the ratios of the sides of a right-angled triangle or by considering the coordinates of points on a unit circle.
The Right-Angled Triangle Definition
Consider a right-angled triangle with one angle denoted by $\theta$.
- The side opposite to the angle $\theta$ is called the Opposite side.
- The side adjacent to the angle $\theta$ (but not the hypotenuse) is called the Adjacent side.
- The longest side, opposite the right angle, is the Hypotenuse.
- Sine ($\sin \theta$): The ratio of the length of the Opposite side to the length of the Hypotenuse. $$ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} $$
- Cosine ($\cos \theta$): The ratio of the length of the Adjacent side to the length of the Hypotenuse. $$ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} $$
- Tangent ($\tan \theta$): The ratio of the length of the Opposite side to the length of the Adjacent side. It can also be expressed as the ratio of sine to cosine: $\tan \theta = \frac{\sin \theta}{\cos \theta}$. $$ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} $$
The Unit Circle Definition
Trigonometric functions can also be defined for any angle, not just those in a right-angled triangle, using the unit circle (a circle with radius 1 centered at the origin of a coordinate plane). For an angle $\theta$ measured counterclockwise from the positive x-axis, a point $(x, y)$ on the unit circle is reached.
- Cosine ($\cos \theta$): The x-coordinate of the point $(x, y)$.
- Sine ($\sin \theta$): The y-coordinate of the point $(x, y)$.
- Tangent ($\tan \theta$): The ratio of the y-coordinate to the x-coordinate ($\frac{y}{x}$).
Units: Degrees vs. Radians
Angles in trigonometry can be measured in two common units:
- Degrees: A full circle is 360 degrees (°). A right angle is 90°.
- Radians: A full circle is $2\pi$ radians. A straight angle is $\pi$ radians. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius.
Applications of Trigonometric Functions
Sine, cosine, and tangent are foundational to many fields:
- Physics: Analyzing wave motion (sound, light, AC circuits), simple harmonic motion, projectile motion, and oscillations.
- Engineering: Structural analysis, electrical engineering (AC circuits), signal processing, robotics, and mechanics.
- Mathematics: Calculus (differentiation and integration of trigonometric functions), geometry, and solving complex equations.
- Navigation: Calculating distances and positions using celestial bodies or GPS coordinates.
- Computer Graphics: Creating rotations, animations, and simulating physical phenomena.
- Surveying: Measuring distances and heights indirectly.
Example Calculation
Let's calculate the sine, cosine, and tangent of an angle of 45 degrees.
Input:
Angle: 45
Unit: Degrees
Calculation:
JavaScript's Math.sin(), Math.cos(), and Math.tan() functions expect angles in radians.
Convert 45 degrees to radians: $ 45 \times \frac{\pi}{180} = \frac{\pi}{4} $ radians.
$ \sin(45^\circ) = \sin(\frac{\pi}{4}) \approx 0.7071 $ (which is $ \frac{\sqrt{2}}{2} $)
$ \cos(45^\circ) = \cos(\frac{\pi}{4}) \approx 0.7071 $ (which is $ \frac{\sqrt{2}}{2} $)
$ \tan(45^\circ) = \tan(\frac{\pi}{4}) \approx 1 $
Angle: 45
Unit: Degrees
Calculation:
JavaScript's Math.sin(), Math.cos(), and Math.tan() functions expect angles in radians.
Convert 45 degrees to radians: $ 45 \times \frac{\pi}{180} = \frac{\pi}{4} $ radians.
$ \sin(45^\circ) = \sin(\frac{\pi}{4}) \approx 0.7071 $ (which is $ \frac{\sqrt{2}}{2} $)
$ \cos(45^\circ) = \cos(\frac{\pi}{4}) \approx 0.7071 $ (which is $ \frac{\sqrt{2}}{2} $)
$ \tan(45^\circ) = \tan(\frac{\pi}{4}) \approx 1 $
The calculator below will perform these calculations for you.