Trigonometry Calculator
Calculate trigonometric and inverse trigonometric functions. Optionally, compute the hypotenuse of a right triangle given two sides.
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About Trigonometric Functions: Trigonometry studies angles and the relationships between triangle sides. Functions like sine, cosine, tangent, and their inverses are used in math, physics, and engineering.
Practical Applications: Useful for students, engineers, architects, and anyone working with angles or right triangles.
Understanding SOH-CAH-TOA
The mnemonic SOH-CAH-TOA is the easiest way to remember the three primary trigonometric ratios for a right triangle. Each part of the mnemonic connects a trigonometric function to a ratio of two sides of the triangle relative to a given angle.
SOH: sin(angle) = Opposite / Hypotenuse
CAH: cos(angle) = Adjacent / Hypotenuse
TOA: tan(angle) = Opposite / Adjacent
The remaining three functions -- cosecant (csc), secant (sec), and cotangent (cot) -- are simply the reciprocals of sine, cosine, and tangent, respectively. For example, csc(angle) = 1 / sin(angle).
The Unit Circle and Radians vs. Degrees
The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It provides a visual framework for understanding all trigonometric functions, not just those in right triangles. Any point on the unit circle can be described as (cos(theta), sin(theta)), where theta is the angle measured from the positive x-axis.
Angles can be measured in degrees or radians. A full rotation is 360 degrees or 2pi radians. The conversion formulas are:
Radians = Degrees x (pi / 180)
Degrees = Radians x (180 / pi)
Radians are the standard unit in calculus and most scientific applications because they simplify many formulas. For instance, the derivative of sin(x) is cos(x) only when x is in radians.
Trigonometric Values for Common Angles
The following table shows the exact values of sine, cosine, and tangent for the most frequently encountered angles. Memorizing these values is extremely helpful for exams and quick mental calculations.
| Degrees | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 0 |
| 30 | pi/6 | 1/2 | sqrt(3)/2 | 1/sqrt(3) |
| 45 | pi/4 | sqrt(2)/2 | sqrt(2)/2 | 1 |
| 60 | pi/3 | sqrt(3)/2 | 1/2 | sqrt(3) |
| 90 | pi/2 | 1 | 0 | Undefined |
Inverse Trigonometric Functions and Applications
Inverse trigonometric functions (arcsin, arccos, arctan) work in the opposite direction of their standard counterparts. Instead of taking an angle and returning a ratio, they take a ratio and return an angle. For instance, arcsin(0.5) = 30 degrees because sin(30 degrees) = 0.5.
Keep in mind that inverse trig functions have restricted ranges to ensure they return a single value: arcsin and arctan return values between -90 and 90 degrees, while arccos returns values between 0 and 180 degrees.
Trigonometry has countless real-world applications. Engineers use it to calculate forces on structures and design bridges. Navigators rely on it for determining positions and plotting courses. In physics, trigonometric functions describe wave motion, oscillations, and alternating current circuits. Computer graphics use trig functions to rotate and transform objects on screen, and surveyors use angles and trigonometric ratios to measure distances and elevations accurately.