Sine and Cosine

$\sin(x)$ and $\cos(x)$ are periodic functions with a period of 360° (or $2\pi$ radians).

Both sine and cosine graphs oscillate between -1 and 1. The sine graph starts at (0,0) while the cosine graph starts at (0,1).

Tangent

$tan(x) = \frac{\sin(x)}{\cos(x)}$

Tangent graphs have vertical asymptotes where cosine is equal to zero (i.e. at 90° and 270°). The graph oscillates between negative and positive infinity.

Inverse Functions

The inverse functions of sine, cosine and tangent are used to calculate angles from given ratios.

When using an inverse function, the input becomes the output, the output becomes of the input.

a(rc)cos vs $\sin^{-1}$

Since these functions can have powers, to avoid confusion we use “arc” to denote the inverse functions of sine, cosine and tangent.

Function	Informal Inverse Function	Formal Inverse Function
$\sin$	$\sin^{-1}$	$\text{asin}$ or $\arcsin$
$\cos$	$\cos^{-1}$	$\text{acos}$ or $\arccos$
$\tan$	$\tan^{-1}$	$\text{atan}$ or $\arctan$

For every x there is one y

Sine, cosine and tangent functions are all functions, meaning that for every x input there is only one y output. However, the inverse functions are not functions since for every y input there are multiple x outputs.

When using the inverse functions the calculator will return the closest value to 0 and positive.

Finding the second answer

CAST Method

To find the second answer for sine and cosine, we can use the CAST method which is not only faster but also:

reduces the chance of making mistakes
works better for more complex problems

In tutor notes.

Sine, Cosine and Tangent Graphs