Trig Identities and Approximations

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Trig Identities and Approximations Revision

Trig Identities and Approximations

From our first identity $\textcolor{blue}{\sin ^2 \theta} + \textcolor{limegreen}{\cos ^2 \theta} \equiv 1$ , we have two new identities.

Make sure you are happy with the following topics before continuing.

Two New Identities

So, we started out in the Basic Trig Identities section by forming the identity $\textcolor{blue}{\sin ^2 \theta} + \textcolor{limegreen}{\cos ^2 \theta} \equiv 1$ . From there, we can derive two new ones.

Identity 1:

$\dfrac{\textcolor{blue}{\sin ^2 \theta} + \textcolor{limegreen}{\cos ^2 \theta} \equiv 1}{\textcolor{blue}{\sin ^2 \theta}}$

gives

$\dfrac{\textcolor{blue}{\sin ^2 \theta}}{\textcolor{blue}{\sin ^2 \theta}} + \dfrac{\textcolor{limegreen}{\cos ^2 \theta}}{\textcolor{blue}{\sin ^2 \theta}} \equiv \dfrac{1}{\textcolor{blue}{\sin ^2 \theta}}$

or,

$1 + \cot ^2 \theta \equiv \cosec ^2 \theta$

Identity 2:

$\dfrac{\textcolor{blue}{\sin ^2 \theta} + \textcolor{limegreen}{\cos ^2 \theta} \equiv 1}{\textcolor{limegreen}{\cos ^2 \theta}}$

gives

$\dfrac{\textcolor{blue}{\sin ^2 \theta}}{\textcolor{limegreen}{\cos ^2 \theta}} + \dfrac{\textcolor{limegreen}{\cos ^2 \theta}}{\textcolor{limegreen}{\cos ^2 \theta}} \equiv \dfrac{1}{\textcolor{limegreen}{\cos ^2 \theta}}$

or,

$\textcolor{red}{\tan ^2 \theta} + 1 \equiv \sec ^2 \theta$

Small Angle Approximations

We also have approximations for $\textcolor{blue}{\sin \theta}$ , $\textcolor{limegreen}{\cos \theta}$ and $\textcolor{red}{\tan \theta}$ is small (i.e. smaller than $1$ , in radians):

$\textcolor{blue}{\sin \theta} \approx \theta$

$\textcolor{limegreen}{\cos \theta} \approx 1 - \dfrac{1}{2}\theta ^2$

$\textcolor{red}{\tan \theta} \approx \theta$

We can use these approximations to find rough values for complicated functions, in order to simplify them.

The expressions do extend to multiples of $\theta$ , also, but only when the product is less than $1$ .

So, our expressions are:

$\textcolor{blue}{\sin n\theta} \approx n\theta$

$\textcolor{limegreen}{\cos n\theta} \approx 1 - \dfrac{1}{2}(n\theta) ^2$

$\textcolor{red}{\tan n\theta} \approx n\theta$

Example 1: Using Approximations

For small values of $\theta$ , find an approximation for $\dfrac{1}{2}\textcolor{blue}{\sin \theta} + 2 \textcolor{limegreen}{\cos \theta} - 2$ , and find any value of $\theta$ where the expression is $0$ .

[3 marks]

The expression $\dfrac{1}{2}\textcolor{blue}{\sin \theta} + 2 \textcolor{limegreen}{\cos \theta} - 2$ can be replaced by our small angle approximations to

$\dfrac{1}{2}\theta + 2 - \theta ^2 - 2$

which can be simplified to

$\dfrac{1}{2}\theta - \theta ^2$

When this expression is equal to zero, we have $\theta ^2 - \dfrac{1}{2}\theta = 0$ , which has roots $\theta = 0, \dfrac{1}{2}$ .

Example 2: Using Identities

Solve $2\sec ^2 \theta = 1 + 3\textcolor{red}{\tan ^2 \theta}$ for all values $-360° \leq x \leq 360°$ .

[3 marks]

First, we need to convert all trig functions to be of one form.

So,

$2 + 2\textcolor{red}{\tan ^2 \theta} = 1 + 3\textcolor{red}{\tan ^2 \theta}$

rearranges to

$1 = \textcolor{red}{\tan ^2 \theta}$

so,

$\textcolor{red}{\tan \theta} = ±1$

or,

$\theta = -315°, -225°, -135°, -45°, 45°, 135°, 225°, 315°$

Trig Identities and Approximations Example Questions

Question 1: Use small angle approximations to find a value for $\cos \dfrac{\pi}{4}$ , the form $a + b\pi ^2$ .

[2 marks]

\cos \theta = 1 - \dfrac{1}{2}\theta ^2

So, $\cos \dfrac{\pi}{4} = 1 - \left( \dfrac{1}{2} \times \dfrac{\pi ^2}{16}\right) = 1 - \dfrac{\pi ^2}{32}$

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Question 2: Find an approximation to $\sin (3\theta ) + 2\cos \theta$ . Use your result to find an approximate value for $\sin 0.6 + 2\cos 0.2$ .

What is the maximum value of $\theta$ where the approximation is accurate?

[3 marks]

\sin (3\theta ) + 2\cos \theta

= 3\theta + 2 - \theta ^2

Using a value of $\theta = 0.2$ , we have $= 3\theta + 2 - \theta ^2 = (3 \times 0.2) + 2 - 0.2^2 = 0.6 + 2 - 0.04 = 2.56$

The maximum value of $\theta$ is $\dfrac{1}{3}$ .

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Question 3: Prove that $\cosec ^2 \theta \equiv 1 + \cot ^2 \theta$ and $\sec ^2 \theta = 1 + \tan ^2 \theta$ .

[4 marks]

$\sin ^2 \theta + \cos ^2 \theta \equiv 1$

$\dfrac{\sin ^2 \theta}{\sin ^2 \theta} + \dfrac{\cos ^2 \theta}{\sin ^2 \theta} \equiv \dfrac{1}{\sin ^2 \theta}$

meaning

$\cosec ^2 \theta \equiv 1 + \cot ^2 \theta$

$\sin ^2 \theta + \cos ^2 \theta \equiv 1$

$\dfrac{\sin ^2 \theta}{\cos ^2 \theta} + \dfrac{\cos ^2 \theta}{\cos ^2 \theta} \equiv \dfrac{1}{\cos ^2 \theta}$

meaning

$\sec ^2 \theta \equiv 1 + \tan ^2 \theta$

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Question 4: Show that $\tan ^4 \theta - \sec ^2 \theta = 5$ can be written as $\sec^4 \theta - 3\sec ^2 \theta - 4 = 0$ and find its solutions for $0° \leq \theta \leq 360°$ .

[5 marks]

$\tan ^4 \theta - \sec ^2 \theta = 5$

$(\sec ^2 \theta - 1)^2 - \sec ^2 \theta = 5$

$\sec ^4 \theta - 3\sec ^2 \theta - 4 = 0$

Let $x = \sec ^2 \theta$ .

Then we have

$x^2 - 3x - 4 = 0$

which has solutions $x = -1, 4$ .

Since $\sec ^2 \theta \neq -1$ , we have $\sec ^2 \theta = 4$ , or $\sec \theta = ±2$ .

$\sec \theta = ±2$ is equivalent to $\cos \theta = ±\dfrac{1}{2}$

So, we have

$\theta = 60°, 120°, 240°, 300°$