Unit 3 - Trigonometry
Required Prior Knowledge
Questions
In the following, give your answers as fractions.
a) Given \(\theta\) is acute and \(\sin\theta=\frac{12}{13}\) determine the value of \(\cos\theta\)
b) Given \(\theta\) is obtuse and \(\sin\theta=\frac{7}{25}\) determine the values of \(\cos\theta\) and \(\tan\theta\)
c) Given \(\theta\) is obtuse and \(\tan\theta=-\frac{8}{15}\) determine the value of \(\sin\theta\)
Solutions
Get Ready (HL)
Questions
Use the compound angle formulae to expand and simplify these expressions:
a) \(\sin\left(\theta + \theta\right)\)
b) \(\cos\left(\theta + \theta\right)\)
c) \(\tan\left(\theta + \theta\right)\)
Solutions
Notes
The double angle identities are:$$\begin{align}\cos\left(2\theta\right)&\equiv\cos^{2}\theta + \sin^{2}\theta\\&\equiv 1-2sin^{2}\theta\\&\equiv 2\cos^{2}\theta - 1\end{align}$$ $$\sin\left(2\theta\right)\equiv 2\sin\theta\cos\theta$$ $$\tan\left(2\theta\right)\equiv \frac{2\tan\theta}{1-\tan^{2}\theta}\qquad\text{(HL)}$$
Examples and Your Turns
Example
Given that \(\sin\alpha =\frac{5}{13}\) where \(\frac{\pi}{2}\lt\alpha\lt\pi\), find the exact value of \(\sin 2\alpha\).
Your Turn
If \(\sin\alpha =-\frac{2}{5}\) and \(\pi\lt\alpha\lt\frac{3\pi}{2}\), find the exact value of \(\sin 2\alpha\) and \(\cos 4\alpha\).
Your Turn
Given that \(\cos\theta =\frac{3}{5}\) and \(\theta\) is in the fourth quadrant, find the exact value of \(\sin 2\theta\).
Your Turn
If \(\alpha\) is acute and \(\cos 2\alpha =\frac{3}{4}\) find the exact values of
a) \(\cos\alpha\)
b) \(\sin\alpha\)
Your Turn
Show that$$\sin(15^{\circ}) = \sqrt{\frac{2 - \sqrt{3}}{4}}$$
Your Turn
Show that$$\sin 3A = 3 \sin A - 4 \sin^3 A$$
Your Turn
Prove the identity$$\frac{\sin(2x)}{1 + \cos(2x)} = \tan x$$
Your Turn
Prove that$$\frac{\sin 2x}{\sin x} = 2 \cos x$$
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
Explain how the different forms of the double angle identity for \(\cos 2\theta\) are useful in different scenarios.
How can you show that the three different forms of the double angle identity for \(\cos 2\theta\) equivalent?
Common Mistakes / Misconceptions
Using the wrong version of the double angle formula for \(\cos 2\theta\).
Forgetting the signs depending on which quadrant the angle is in.
Connecting This to Other Skills
These results are directly derived from the Compound Angle Formulae (3.13) and builds upon the ideas of Trigonometric Identities (3.12).
We will use the double angle identities to Solve Trigonometric Equations (3.16).
The double angle identities are used regularly when integrating trigonometric functions (5.6).
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?