Trigonometric identities are identities in mathematics that involve trigonometric functions such as sin(x), cos(x) and tan(x). Identities, as opposed to equations, are statements where the left hand side is equivalent to the right hand side. A \equiv symbol, which means ‘equivalent’, is used instead of the usual ‘equals’ sign. Equations can be solved to find values of x, for instance. An identity, however, cannot be solved, no additional information has been given. They can, however, be used to solve equations –  see Trigonometric Equations. It follows that Trigonometric Identities are simply equivalent trigonometric expressions.


Basic Trigonometric Identities

You may be required to memorise some or all of the following basic trigonometric identities and common values:

Fundamental Identities

\frac{\sin(\theta)}{\cos(\theta)}\equiv\tan(\theta)
\cos^2(\theta)+\sin^2(\theta)\equiv 1.


\begin{array}{l}\sin(-\theta)\equiv-\sin(\theta)\\\cos(-\theta)\equiv\cos(\theta)\\\sin(90^\circ-\theta)\equiv\cos(\theta)\\\cos(90^\circ-\theta)\equiv\sin(\theta)\\\sin(180^\circ-\theta)\equiv\sin(\theta)\\\cos(180^\circ-\theta)\equiv-\cos(\theta)\\\end{array}

Trigonometric Ratios

\begin{array}{c}\sin(30^\circ)=\frac{1}{2}\\\cos(30^\circ)=\frac{\sqrt{3}}{2}\\\tan(30^\circ)=\frac{\sqrt{3}}{3}\\\\\sin(45^\circ)=\frac{\sqrt{2}}{2}\\\cos(45^\circ)=\frac{\sqrt{2}}{2}\\\tan(45^\circ)=1\\\\\sin(60^\circ)=\frac{\sqrt{3}}{2}\\\cos(60^\circ)=\frac{1}{2}\\\tan(60^\circ)=\sqrt{3}\end{array}


Where do these Trigonometric Identities come from?

Fundamental Identities

Firstly, the identity \tan(\theta)\equiv\frac{\sin(\theta)}{\cos(\theta)} can be seen using SOHCAHTOA. Recall that \sin(\theta)=\frac{OPP}{HYP} and  \cos(\theta)=\frac{ADH}{HYP}. It follows that

\tan(\theta)=\frac{OPP}{ADJ}=\frac{OPP}{HYP}\div \frac{ADJ}{HYP}=\frac{\sin(\theta)}{\cos(\theta)}

trigonometric identitiesSecondly, recall that \cos(\theta) and \sin(\theta) can be thought of as the x and y coordinates of a particle traversing a unit circle. It follows that the identity \cos^2(\theta)+\sin^2(\theta)\equiv 1 results from an application of Pythagoras. See Trigonometric Graphs.


Now consider \sin(-\theta)\equiv-\sin(\theta), for example. This type of identity can be seen by performing transformations to standard curves. See Transformations.

Firstly, consider the graph of \sin(-\theta). It is obtained by replacing  \theta with -\theta. That is, a reflection in the y-axis. This is precisely the sin curve multiplied by -1, i.e. reflected across the x-axis. This is the above identity, as expected. Functions with this property are known as odd functions. The other identities can be derived in a similar way. Beware – it is common for student to perform composite transformations incorrectly.

Click here to see other trigonometric graph transformations.

Trigonometric Ratios

The Trigonometric Ratios seen above can be found without using a calculator. They can be found from two ‘special triangles’:

Firstly, the triangle on the left is an equilateral triangle. As you can see, all of the sides have length 2. This means that all of the angles are 60 degrees. If this triangle is split down the middle, then each of the angles at the top is 30 degrees. Note that the length of this vertical line is root 3 and it follows from Pythagoras. The sin, cos and tan values for 30 and 60 degrees can be found from this triangle using SOHCAHTOA.  For example, \cos(30^\circ)=\frac{ADJ}{HYP}=\frac{\sqrt{3}}{2}.

Secondly, the triangle on the right is an isosceles triangle. As you can see, two of the sides have length 1. It follows from Pythagoras that the length of the hypoteneuse is root 2. Since one of the angles is a right angle, the remaining angles are both 45 degrees. The sin, cos and tan values for 45 degrees can be found from this triangle also using SOHCAHTOA. For example, \sin(45^\circ)=\frac{OPP}{HYP}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}.

In more advanced trigonometry, other trigonometric values may also be found. For example, it is possible to find \tan(15^\circ) using the compound angle identities below. This can be achieved by choosing \alpha=45^\circ and \beta=30^\circ, for instance. More on this soon.


Trigonometric Identities Examples

Example 1

Simplify

\frac{\sqrt{1-\cos^2(x)}}{\cos(x)}

Example 2

Show that \frac{\sin^4(\theta)-\cos^4(\theta)}{\cos^2(\theta)}\equiv\tan^2(\theta)-1.


Example 3

Given that \theta is obtuse and \cos(\theta)=-\frac{2}{5}, find the value of \tan(\theta).


Example 4

Given that x=2\cos(\alpha) and y=3\sin(\alpha), find the value of 9x^2+4y^2.



More Trigonometric Identities

In addition to the above, you may be require to master the following identities. These identities are more complicated than the one seem above and so may feature later on in your course.

Double Angle Identities

\begin{array}{l}\sin(2\theta)\equiv2\sin(\theta)\cos(\theta)\\\cos(2\theta)\equiv\cos(\theta)^2-\sin^2(\theta)\\\cos(2\theta)\equiv2\cos^2(\theta)-1\\\cos(2\theta)\equiv1-2\sin^2(\theta)\end{array}

Compound Angle Indentities

\begin{array}{l}\sin(\alpha\pm\beta)\equiv\sin(\alpha)\cos(\beta)\pm\cos(\alpha)\sin(\beta)\\\cos(\alpha\pm\beta)\equiv\sin(\alpha)\sin(\beta)\mp\cos(\alpha)\cos(\beta)\\\tan(\alpha\pm\beta)\equiv\frac{\tan(\alpha)\pm\tan(\beta)}{1\mp\tan(\alpha)\tan(\beta)}\end{array}


For more trigonometric identities visit Wikipedia Trig. Identities.


Also see other Things to Remember.


Alternatively, click here to find Questions by Topic and scroll down to all past TRIGONOMETRY questions to practice some more.


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