Lesson 6.1: Trigonometric identities and equations

Introduction

Welcome to Lesson 6.1 of Foundation Further Mathematics! In this lesson, we will dive deep into the world of trigonometric identities and equations. By the end of this lesson, you will not only understand the various identities but also know how to solve trigonometric equations over specified intervals.

Learning Objectives:

• Understand and prove compound-angle, double-angle, and half-angle identities.
• Learn the R-form equation: $a \cos \theta + b \sin \theta = R \cos(\theta \mp \alpha)$ and its applications.
• Solve trigonometric equations over a specified interval, including equations that can be transformed into a quadratic form.
• Familiarize yourself with reciprocal (sec, cosec, cot) and inverse trigonometric functions along with their important identities.
• Prove and apply both compound and double-angle identities.

Trigonometric Identities

Compound Angle Identities

Compound angle identities help us express the sine and cosine of a sum or difference of two angles. Here are the key identities:

• For sine:

$$ \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b $$

• For cosine:

$$ \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b $$

Double Angle Identities

Double angle identities are a specific case of compound angle identities where both angles are the same:

• For sine:

$$ \sin(2\theta) = 2 \sin \theta \cos \theta $$

• For cosine:

$$ \cos(2\theta) = \cos^2 \theta - \sin^2 \theta $$

or equivalently,

$$ \cos(2\theta) = 2\cos^2 \theta - 1 $$

or even,

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

Half Angle Identities

Half angle identities allow us to find the sine and cosine of half angles:

• For sine:

$$ \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}} $$

• For cosine:

$$ \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}} $$

R-Form Identity

The R-form identity is useful for rewriting expressions involving sine and cosine:

$$ a \cos \theta + b \sin \theta = R \cos(\theta \mp \alpha) $$

Where:

• $R = \sqrt{a^2 + b^2}$
• $\tan \alpha = \frac{b}{a}$

Example

Let's find the R-form for the expression $3 \cos \theta + 4 \sin \theta$:

1. Calculate $R$:

$$ R = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$

1. Calculate $\alpha$:

$$ \tan \alpha = \frac{4}{3} \Rightarrow \alpha = \tan^{-1}\left(\frac{4}{3}\right) $$

1. Rewrite the expression:

$$ 3 \cos \theta + 4 \sin \theta = 5 \cos\left(\theta - \alpha\right) $$

Solving Trigonometric Equations

Solving Basic Trigonometric Equations

To solve trigonometric equations, we often set the equation equal to a known value, usually in the range of $[-1, 1]$ for sine and cosine functions:

Example

Solve $\sin \theta = \frac{1}{2}$ for $\theta$ in the interval $[0, 2\pi]$:

1. Using the unit circle, we find the angles where sine is $\frac{1}{2}$:

• $\theta = \frac{\pi}{6}$ and $\theta = \frac{5\pi}{6}$.

1. Thus, the solutions are:

$$ \theta = \frac{\pi}{6}, \frac{5\pi}{6} $$

Solving Quadratic Equations in Trigonometric Form

Some trigonometric equations can be transformed into quadratic forms, making them easier to solve:

Example

Solve $\sin^2 \theta - \sin \theta - 2 = 0$:

1. Let $x = \sin \theta$. Then, we rewrite the equation:

$$ x^2 - x - 2 = 0 $$

1. Factoring gives:

$$ (x - 2)(x + 1) = 0 $$

1. Thus, $x = 2$ or $x = -1$. The values of sine must lie between $[-1, 1]$. Therefore:

$$ \sin \theta = -1 \Rightarrow \theta = \frac{3\pi}{2} $$

Inverse and Reciprocal Functions

Key Identities

The key reciprocal identities are:

• $\sec \theta = \frac{1}{\cos \theta}$
• $\csc \theta = \frac{1}{\sin \theta}$
• $\cot \theta = \frac{1}{\tan \theta}$

Inverse Trigonometric Functions

The primary inverse functions are:

• $\sin^{-1}(x)$ (arcsine)
• $\cos^{-1}(x)$ (arccosine)
• $\tan^{-1}(x)$ (arctangent)

Example

To find the angle $\theta$ if $\sin \theta = x$, we use:

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

Conclusion

In this lesson, we extensively covered trigonometric identities and equations. Mastering these concepts is essential for your journey through calculus and higher-level mathematics. Make sure to practice rewriting angles and solving equations, as these skills will prove invaluable!

Study Notes

• Compound-angle identities relate the sum or difference of angles.
• Double angle identities simplify calculations for angles multiplied by 2.
• R-form rewrites combinations of sine and cosine into a single cosine function.
• Trigonometric equations can often be converted to simpler quadratic forms.
• Remember reciprocal and inverse trigonometric functions and their properties.