Unit 3 - Trigonometry
Required Prior Knowledge
Questions
Given a right-angled triangle with sides \(3\) and \(7\), find the length of the hypotenuse.
How are the values of \(\sin \theta\) and \(\cos\theta\) related to the unit circle?
Solutions
Get Ready
Questions
Using the unit circle definitions of \(\sin\theta\) and \(\cos\theta\), find a formula that connects them.
Solutions
Notes
The Pythagorean Identity is $$\sin^{2}\theta + \cos^{2}\theta \equiv 1$$We also know the identity$$\frac{\sin\theta}{\cos\theta}\equiv \tan\theta$$We can use these identities, or use Pythagoras’ Theorem with the unit circle, to determine one trigonometric ratio if we know another.
Examples and Your Turns
Example
If \(\tan x =\frac{3}{4}\) and \(\pi\lt x\lt\frac{3\pi}{2}\) find \(\sin x\) and \(\cos x\).
Your Turn
If \(\cos \theta =\frac{1}{3}\) and \(\frac{3\pi}{2}\lt\theta\lt 2\pi\) find \(\sin\theta\) and \(\tan\theta\).
Your Turn
Given that \(\sin \theta =\frac{5}{13}\) and \(\frac{\pi}{2}\lt\theta\lt\pi\) find \(\cos \theta\) and \(\tan \theta\).
Your Turn
Given that \(\sin\theta = \frac{2}{3}\) and \(0\lt\theta\lt 2\pi\) find all possible values of \(\cos \theta\) and \(\tan\theta\).
Your Turn
Given that \(\tan \theta =-2\) and \(\frac{3\pi}{2}\lt\theta\lt 2\pi\) find \(\cos \theta\) and \(\sin\theta\).
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
Why do we need to be told the quadrant of the angle to be able to find the trigonometric ratio?
Common Mistakes / Misconceptions
The most common mistake is to forget to think about the quadrant, and hence the sign of the answer.
Connecting This to Other Skills
This builds upon the ideas of the Unit Circle (3.5).
We will meet more Trigonometric Identities (3.12) and use these to solve Trigonometric Equations (3.16).
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?