Unit 3 - Trigonometry
Required Prior Knowledge
Questions
Consider the unit fractions$$\begin{array} &\frac{1}{6}&\frac{1}{4}&\frac{1}{3}&\frac{1}{2}\end{array}$$Using these fractions, and only adding and subtracting, make the following fractions
a) \(\frac{2}{3}\)
b) \(\frac{1}{12}\)
c) \(\frac{5}{12}\)
Solutions
Get Ready
Questions
Complete this table for angles \(A\) and \(B\), including some angles of your own.
What do you notice? What do you wonder?
Investigate some other trigonometric functions of sums and differences of angles.
| \(A\) | \(B\) | \(\cos A\) | \(\cos B\) | \(\cos(A - B)\) | \(\cos A - \cos B\) | \(\cos A \cos B + \sin A \sin B\) |
|---|---|---|---|---|---|---|
| \(47^{\circ}\) | \(24^{\circ}\) |
\(0.6820\)
?
|
\(0.9135\)
?
|
\(0.9336\)
?
|
\(-0.2315\)
?
|
\(0.9336\)
?
|
| \(138^{\circ}\) | \(49^{\circ}\) |
\(-0.7431\)
?
|
\(0.6561\)
?
|
\(-0.3584\)
?
|
\(-1.3992\)
?
|
\(-0.3584\)
?
|
| \(3\) | \(2\) |
\(-0.9900\)
?
|
\(-0.4161\)
?
|
\(0.5403\)
?
|
\(-0.5739\)
?
|
\(0.5403\)
?
|
Notes
The Compound Angle Formulae are$$\cos\left(A\pm B\right)\equiv \cos A \cos B \mp \sin A \sin B$$ $$\sin\left(A\pm B\right)\equiv \sin A \cos B \pm \sin B \cos A$$ $$\tan\left(A\pm B\right)\equiv \frac{\tan A + \tan B}{1\mp \tan A \tan B}$$
Proofs
Examples and Your Turns
Example
Show that $$\cos \left(\frac{\pi}{12}\right) = \frac{\sqrt{3} + 1}{2\sqrt{2}}$$
Your Turn
Find the exact value of $$\sin\left(\frac{7\pi}{12}\right)$$
Example
Given that \(\sin\theta = \frac{3}{5}\) and \(\sin\phi = \frac{12}{13}\), where both \(\theta\) and \(\phi\) are acute angles, find the exact value of \(\cos\left(\theta - \phi\right)\).
Your Turn
Given that \(\sin A = \frac{4}{5}\) and \(\cos B = \frac{12}{13}\), where both \(A\) and \(B\) are acute angles, find the exact value of \(\sin\left(A - B\right)\).
Your Turn
Given that \(\sin x = \frac{1}{3}\) and \(\sin y = \frac{1}{2}\), where both \(x\) and \(y\) are acute angles, find the exact value of \(\sin\left(x+y\right)\).
Your Turn
Show that$$\cos(A + B)\cos(A - B) \equiv \cos^2 A - \sin^2 B$$
Your Turn
Prove the identity$$\frac{\cos(A - B)}{\cos(A)\sin(B)} = \cot(B) + \tan(A)$$
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 can we use the compound angle formulae to determine new exact values?
Common Mistakes / Misconceptions
A common mistake is to get the signs wrong, particularly in the formula for \(\cos\left(A+B\right)\).
Connecting This to Other Skills
You often have to use the Exact Values (3.6) in answering questions involving compound angle formulae.
We can prove new Trigonometric Identities (3.12) using the compound angle formulae.
A direct consequence of the compound angle formulae are the Double Angle Formulae (3.14).
In Differentiating Trig Functions (4.12) we will use the compound angle formulae to prove the derivatives of \(\sin x\) and \(\cos x\).
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?