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# The Sine Graph

### The Sine Graph

The sine graph comes from trigonometry, and produces a distinctive wave shape which can be found in waves of sound and light.

The Sine function appears in the study of geometry in its connection with right angled triangles. We can plot how a sine function changes as its angle changes on a graph.

The Sine relationship is defined, relative to an angle, as:

$sin(x) = \dfrac{opposite}{hypotenuse}$

where $x$ is the angle in degrees.

If you can't remember which sides of a triangle are known as opposite and hypotenuse, check out our lesson on Sine, Cosine and Tangent.

If we know either the length of the opposite side or the length of the hypotenuse, and the angle, we are able to work out the unknown length with the above equation.

For $y = sinx$ where $0 < x < 360$, the graph has a maximum when x is at what value?

The sine curve is symmetrical about $x=90$, and repeats every $360\degree$. We can use the symmetry to identify similar points on the curve.

We can work out the size of an angle, and therefore the location on the x-axis, using the function $sin^{-1}$. For example:

$sin^{-1}(1)=90\degree$

is the inverse of:

$sin(90\degree) = 1$

You can calculate the inverse sine using the graph above or a calculator.

What is $sin^{-1}(0.5)$?

For $y = sinx$ between 0 and 360, $sin14 = 0.24$. Find another value of x that would give 0.24