Compound and Double Angle Formula

First of all its important to know what sin and cos are. You probably remember using trigonometry to find either lengths or angles of triangles. When calculating sin and cos functions we model them using the unit circle, a circle of a radius of one.

Unit Circle¹

θ is the angle measured anticlockwise from the positive x axis. In the unit circle sin(θ) measures the distance along the y axis and cos(θ) the distance along the x axis.

As θ completes a full rotation (360 degrees or 2π), then the values of sin(θ) and cos(θ) will repeat themselves.

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

This can be seen as the gradient of the tangent on the unit circle. However it is only positive in the first and fourth quadrants.

The other important property from the unit circle is if we use Pythagoras’ formula for a hypotenuse:

$sin^{2}(\theta)+cos^{2}(\theta)=1$

Symmetry Properties

There are many symmetry properties which can be found in your text book, so we will just look at the main negative angle symmetry properties.

$cos(-\theta)=cos(\theta)$

$sin(-\theta)=-sin(\theta)$

$tan(-\theta)=\frac{-sin(\theta)}{cos(\theta)}=-tan(\theta)$

Complementary Relationships

The sin(θ) function is just an offset version of the cos(θ). Therefore:

$sin(\frac{\pi}{2}-\theta)=cos(\theta)$

$cos(\frac{\pi}{2}-\theta)=sin(\theta)$

Addition Formula

Now we will look at addition formulas which are used to model the addition of known sin, cos and tan exact values to find the unknown value.

$cos(\theta-\phi)=cos(\theta)cos(\phi)+sin(\theta)sin(\phi)$

$cos(\theta+\phi)=cos(\theta)cos(\phi)-sin(\theta)sin(\phi)$

$cos(\theta+\phi)=sin(\theta)cos(\phi)+cos(\theta)sin(\phi)$

$cos(\theta-\phi)=sin(\theta)cos(\phi)-cos(\theta)sin(\phi)$

$tan(\theta+\phi)=\frac{tan(\theta)+tan(\phi)}{1-tan(\theta)tan(\phi)}$

Double Angle Formula

The double angle formula can be extremely useful especially when differentiating or integrating functions. These are found by substituting the value of φ for θ in the addition formulas above.

$cos(\theta+\theta)=cos(\theta)cos(\theta)-sin(\theta)sin(\theta) \newline cos(2\theta)=cos^{2}(\theta)-sin^{2}(\theta)$

Then we can go further by substituting in the values of sin and cos from:

$sin^{2}(\theta)+cos^{2}(\theta)=1$

$cos(2\theta)=cos^{2}(\theta)-(1-cos^{2}(\theta)) \newline cos(2\theta)=2cos^{2}(\theta)-1$

$cos(2\theta)=(1-sin^{2}(\theta))-sin^{2}(\theta)) \newline cos(2\theta)=1-2sin^{2}(\theta)$

Finally we can do the same with our tan addition formula:

$tan(\theta+\theta)=\frac{2tan(\theta)}{1-tan^{2}(\theta)}$

Example

In this example we will try some function manipulation in order to simplify for differentiation.

$f(x)=x-2xsin^{2}(4x)$

Find f’(x).

Hopefully this looks familiar to on of the double angle formula that we were looking at before:

$cos(2\theta)=1-2sin^{2}(\theta)$

Step 1 Is manipulate the the original equation to get it into this form:

$f(x)=x-2xsin^{2}(4x) \newline =x(1-2sin^{2}(4x))$

Step 2 sub in $cos(8x)=(1-2sin^{2}(4x))$

$f(x)=x(cos(8x))$

Step 3 Differentiate with the product rule

$u=x, u'=1 \newline v=cos(8x), v'=-8sin(8x)$

$f'(x)=uv'+vu' \newline =-8xsin(8x)+cos(8x)$

http://www.sparknotes.com/math/trigonometry/trigonometricfunctions/section5.rhtml [29 sept] ↩

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