Unit 3 - Trigonometry
Required Prior Knowledge
Questions
Using the Unit Circle, complete the following:
a) \(\sin^{2}x+\cos^{2}x=\)
b) \(\sin\left(-x\right)=\)
c) \(\sin\left(x+\frac{\pi}{2}\right)=\)
d) \(\tan x=\)
e) \(\cos\left(\pi - x\right)=\)
f) \(\tan\left(x+\pi\right)=\)
Solutions
Get Ready
Questions
Consider the algebraic statement \(\left(x-2\right)\left(x+2\right)=x^{2}-4\).
a) What is an identity in mathematics?
b) Why is the statement above an identity?
c) In what ways is an identity different from an equation, such as \(\left(x-2\right)\left(x+2\right)=0\)?
Solutions
Notes
An identity is a mathematical statement consisting of two expressions that are equal for all values of the variables.
This differs from an equation which has two expressions that are equal for specific (perhaps infinitely many) solutions.
We use the symbol \(\equiv\) to show an identity.
For example,$$3a+2a\equiv 5a$$$$x^{2}-y^{2}\equiv\left(x-y\right)\left(x+y\right)$$
There are several important trigonometric identities.
For all values of \(\theta\)$$\begin{align}\tan\theta&\equiv\frac{\sin\theta}{\cos\theta}\\ \cos^{2}\theta +\sin^{2}\theta&\equiv 1\\ 1+\tan^{2}\theta&\equiv\sec^{2}\theta\qquad\text{(HL)}\\ \cot^{2}\theta +1&\equiv\operatorname{cosec}^{2}\theta\qquad\text{(HL)}\end{align}$$In order to prove an identity start with the more complicated side and try to simplify it using known trigonometric identities.
True or False
| Equation | Status | Equation | Status |
|---|---|---|---|
| \( \sec x = \frac{1}{\sin x} \) |
False
?
|
\( \sin^2 x - \cos^2 x = 1 \) |
False
?
|
| \( \cos^2 2\theta + \sin^2 2\theta = 1 \) |
True
?
|
\( \frac{2 \sin x}{2 \cos x} = 2 \tan x \) |
False
?
|
| \( \cot x = \frac{\cos x}{\sin x} \) |
True
?
|
\( (\sin x)^2 + (\cos x)^2 = 1 \) |
True
?
|
| \( \tan x \cos x = \sin x \) |
True
?
|
\( \tan^2 x = 1 + \sec^2 x \) |
False
?
|
Examples and Your Turns
Example
Show that$$\frac{1 + \cos \theta}{\sin \theta} + \frac{\sin \theta}{1 + \cos \theta} \equiv \frac{2}{\sin \theta}$$
Your Turn
Prove that$$\frac{\sin x}{1+\cos x}\equiv\frac{1-\cos x}{\sin x}$$
Your Turn
Show that$$1-2\sin^{2}\theta \equiv2\cos^{2}\theta -1$$
Your Turn (HL)
Show that$$\cot\theta +\tan\theta\equiv\operatorname{cosec}\theta\sec\theta$$
Your Turn (HL)
Show that$$\frac{1}{1+\sin\theta}+\frac{1}{1-\sin\theta}\equiv2\sec^{2}\theta$$
Your Turn
Show that$$\tan^{2}\theta\equiv\sin^{2}\theta\left(1+\tan^{2}\theta\right)$$
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
How are the Pythagorean Identities related to the Unit Circle and Pythagoras’ Theorem?
How could you use a graphing calculator to check to see if an identity is likely to be true? Why is this not a formal proof of the identity?
Why do we start with the more complicated side when trying to prove a trigonometric identity?
Common Mistakes / Misconceptions
The most common error is algebraic errors in the process of the proof. Check each step carefully.
Whilst it can be an adequate approach, starting with both sides and trying to show they both equal the same thing can lead to errors like \(1=1\). It is better to start with one side and show how you can manipulate it to get to the other side.
Connecting This to Other Skills
The Unit Circle (3.5) definition of the trigonometric functions leads to the most basic trigonometric identities.
The concept of Proof (1.7) is important in these questions, in particular direct proof working from the Left Hand Side to the Right Hand Side.
The Reciprocal Trig Functions (3.11) appear in several of the trigonometric identities.
When we get to Solving Trigonometric Equations (3.16) we will see that sometimes we need to simplify an equation first, using a Trigonometric Identity.
When Integrating Trig Functions (5.6) we will need to use trigonometric identities, especially the Pythagorean Identity, to help simplify expressions.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?