## Basic Identities

For angles $$A$$ and $$B$$ it can be proven geometrically that

$$\sin ( A + B ) = \sin A \cos B + \cos A \sin B$$ and $$\sin ( A - B ) = \sin A \cos B - \cos A \sin B$$

It can also be geometrically verified that

$$\cos ( A + B ) = \cos A \cos B - \sin A \sin B$$ and $$\cos ( A - B ) = \cos A \cos B + \sin A \sin B$$.

Using these results it can be shown that

$$\tan ( A + B ) = \dfrac{ \tan A + \tan B }{ 1 - \tan A \tan B }$$ and $$\tan ( A - B ) = \dfrac{ \tan A - \tan B}{1 + \tan A \tan B}$$.

These are referred to as the addition and subtraction identities.

## Exercise 5.3.1

Use the exact values of the trigonometric functions of $$30^\circ$$ and $$45^\circ$$ and the addition and subtraction identities to find exact values of the trigonometric functions of $$75^\circ$$ and $$15^\circ$$.

See Solution

## Exercise 5.3.2

Prove the cofunction identity $$\cos ( 90^\circ - A ) = \sin A$$ using the difference identity for cosine.

See Solution

## Exercise 5.3.3

Find the exact value of $$\sin 17^\circ \cos 13^\circ + \cos 17^\circ \sin 13^\circ$$ without using a calculator.

See Solution

## Exercise 5.3.4

Find an identity for $$\tan \left( x + \frac{\pi}{4} \right)$$ in terms of $$\tan x$$.

See Solution

## Linear combinations of sine and cosine

A sum of multiples of two functions is called a linear combination of the two functions.

Thus, if $$f ( x ) = a \sin x + b \cos x$$, then $$f ( x )$$ is a linear combination of $$\sin x$$ and $$\cos x$$.

We are going to see an example of the general principle that every linear combination of $$\sin x$$ and $$\cos x$$ is a sinusoidal function.

Let $$f ( x ) = 3 \sin x + 4 \cos x$$. Show that $$f$$ is a sinusoidal function. Find the amplitude, period and phase shift.

.

The first step is to find the square root of the sum of the squares of the coefficients. In this case, that is$$\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}= 5$$. Factor $$5$$ out of the expression to get $$f ( x ) = 5 \left( \frac{3}{5} \sin x + \frac{4}{5}\cos x\right)$$

Next, regard the coefficient of $$\sin x$$ as the cosine of some angle $$\phi$$ and the coefficient of $$\cos x$$ as the sine of the same angle $$\phi$$. We can make this assumption since the sum of the squares of the coefficients equals $$1$$ as does the sum of the squares of $$\cos\phi$$ and $$\sin\phi$$ for any angle $$\phi$$..

This gives $$f ( x ) = 5 ( \cos\phi \sin x + \sin\phi \cos x)$$

Using the sum formula for sine, this can be re-written $$f ( x ) = 5 \sin ( x + \phi )$$.

The amplitude is $$5$$, the period is $$2 \pi$$ and the phase shift is $$-\phi$$.

What is the value of $$\phi$$ ?

Since both the sine and the cosine of $$\phi$$ are positive, $$\phi$$ is an angle in quadrant I. So we can calculate $$\phi=\arccos \left( \frac{3}{5} \right)$$ or alternately, $$\phi=\arcsin \left( \frac{4}{5} \right)$$. So $$\phi=0.9273$$ radians or $$53.1^\circ$$.

## Exercise 5.3.5

Find the amplitude, period and phase shift of $$f ( x ) = - 2 \sin ( x ) + 3 \cos ( x )$$

Sketch the graph.

See Solution