Trig Identities

For SL there are a small number of identities that you need to be familiar with. The problems that requires them are not always obvious. In general if you're stuck on a trig problem look to the identities for a hint. This is especially true on Paper 1.

### Pythagorean Identity

This one is named appropriately as it is simply an extension of the Pythagorean formula.

(1)
\begin{align} cos^2 \theta + sin^2 \theta = 1 \end{align}

If you see a cosine or sine function squared. This identity should come to mind. Its not always need, but frequently is when a squared trig function is involved.

### Tangent Identity

At some level its a definition, but here it is, its in the data booklet for exams after 2014.

(2)
\begin{align} tan \theta = \frac{sin \theta}{cos \theta} \end{align}

### Double Angle Formulae

If you see a problem with a double angle such as $sin (2 \theta)$ or $cos(2A)$ then these are often needed.

(3)
\begin{align} sin (2 \theta) = 2 sin \theta cos \theta \end{align}

The next identity is written as one line in the data booklet but is essentially written three different ways. Use the one that gets you where you want to go.

(4)
\begin{align} cos (2 \theta) = cos^2 \theta - sin^2 \theta \end{align}

or

(5)
\begin{align} cos (2 \theta) = 2 cos^2 \theta - 1 \end{align}

or

(6)
\begin{align} cos (2 \theta) = 1 - 2 sin^2 \theta \end{align}

### A Comment on Notation

The notation with trig functions is not always awesome. Just to be clear:

(7)
$$cos^2 x = (cos x)^2=(cos x)(cos x)$$

### Example(s)

Example 1

Solve the equation $2cos (x) = sin (2x)$ for $0\leq x \leq 3\pi$

Got another problem you'd like to see solved? Put it in the comments below. I'll add it to the page.