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Sine Function, Cosine Function, and Tangent Function Explained

There are three basic functions to understand in trigonometry: sine function, cosine function, and tangent function. Each function takes in an angle measure, which is called theta and represented by the symbol sine function: symbol θ. The angle measure is usually expressed in radians or degrees. The output of the function is a ratio comparing one side of the triangle to another.

Using the Sine Function in a Right Triangle

Let's use a right triangle as an example.

Sine function in a right triangle

Here, sin C, or the angle labeled C above, is comparing the ratio of the opposite side to the hypotenuse (review the Pythagorean Theorem here). The value of sin C, in this case, is calculated by opposite over hypotenuse formula, or 2.5 over 6 formula. Sines and cosines both have ranges of -1 to 1, meaning that the biggest and smallest y-coordinates of a graph for sin X or cosine X are constrained to -1 to 1.

The domain, or possible values of the x-coordinate, is all real numbers, meaning anywhere on the number line! The tangent function also has a range of all real numbers but has a slightly different domain.

Graphing the Sine Function

You can see that the sine function is a periodic function, meaning that its amplitude is cyclical no matter how far you graph to the right or to the left. You may hear this kind of graph referred to as the "sine wave" or as "sinusoidal." You introduce a phase shift if you add other coefficients and variables to the function. This moves the graphed function up, down, left, or right.

Sine function: Sine wave graph

Other Trigonometric Functions

There are many other trig functions comparing these ratios of a triangle. Use the helpful mnemonic SOHCAHTOA to remember the definitions:

Sine is the opposite side divided by the hypotenuse.

Cosine is the adjacent side divided by the hypotenuse.

Tangent is the opposite side divided by the adjacent side.

Each of these functions also has an inverse function, which means the numerator and denominator in the fraction are switched. The inverse functions work like this:

Secant is the reciprocal of cosine: the hypotenuse divided by the adjacent side.

Cosecant is the reciprocal of sine: the hypotenuse divided by the opposite side.

Cotangent is the reciprocal of tangent: the adjacent side divided by the opposite side.

What About Inverse Sine?

Inverse sine is the function where you are given the measurements of the lengths of the triangle and need to figure out the original angle. You can think of it as reversing the function. For our example from earlier, we can figure out angle C by taking the inverse function.

Inverse sine formula

c = 24.62

Sine function in a right triangle

Learn Sine Function, Cosine Function, and Tangent Function

The sine function, cosine function, and tangent function are the three main trigonometric functions.

The input or domain is the range of possible angles. The output or range is the ratio of the two sides of a triangle. The two sides being compared depend on which function you are using: sine, cosine, tangent, cosecant, secant, or cotangent. The inverse of any of these functions is the function in reverse: taking the lengths of the two sides and computing the angle.

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