Unit 3 - Trigonometry
Required Prior Knowledge
Questions
If we know that \(\sin 73^{\circ} =0.956\) which of these can we work out?
a) \(\sin 107^{\circ} \)
b) \(\sin 163^{\circ}\)
c) \(\cos 73^{\circ}\)
d) \(\cos 17^{\circ}\)
e) \(\sin 287^{\circ}\)
f) \(\sin 433^{\circ}\)
g) \(\cos 163^{\circ}\)
h) \(\cos 377^{\circ}\)
i) \(\sin 253^{\circ}\)
Solutions
Get Ready
Questions
What is an inverse function?
What facts do you know about inverse functions?
Solutions
Notes
The graphs of the trigonometric functions are NOT one-to-one.
Hence, we have to restrict the domain in order to have an inverse function.
Whilst you have probably used the notation \(y=\sin^{-1} x\) for the inverse function of \(y=\sin x\), the formal name for the inverse is \(y=\arcsin x\).
This is to avoid possible confusion between inverses and reciprocals.
$$y=\sin x $$
$$y=\cos x $$
$$y=\tan x $$
$$y=\arcsin x $$
$$y=\arccos x $$
$$y=\arctan x $$
Examples and Your Turns
Example
Evaluate
a) \(\arccos \frac{\sqrt{2}}{2}\)
b) \(\arctan \sqrt{3}\)
Your Turn
Evaluate
a) \(\arcsin \frac{1}{2}\)
b) \(\arccos \frac{\sqrt{3}}{2}\)
c) \(\arctan 1\)
d) \(\arctan \frac{1}{\sqrt{3}}\)
e) \(\arcsin\left(-\frac{\sqrt{3}}{2}\right)\)
f) \(\arccos -1\)
Example
Evaluate$$\sin\left(\arcsin\frac{3\pi}{4}\right)$$
Your Turn
Evaluate$$\arctan\left(\tan\frac{\pi}{4}\right)$$
Example
Evaluate$$\tan\left(\arccos -\frac{3}{5}\right)$$
Your Turn
Evaluate$$\cos\left(\arcsin\frac{12}{13}\right)$$
Example
Evaluate$$\sin \left(\arctan \left(\frac{1}{\sqrt{3}}\right)\right) + \cos \left(\arcsin \left(\frac{1}{\sqrt{2}}\right)\right)$$
Your Turn
Evaluate$$3 \arcsin(3x) = \arcsin \left(\frac{1}{2}\right) + \arccos \left(\frac{1}{2}\right)$$
Your Turn
Write \(\sin\left(\arcsin a + \arccos b \right)\) in terms of \(a\) and \(b\).
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 restrict the domains of the trigonometric functions for them to have an inverse?
Why are the restricted domains different for sine and cosine?
Explain why \(\arcsin\left(\sin 2\pi\right)\) is not equal to \(2\pi\).
Common Mistakes / Misconceptions
Giving a value outside of the restricted domain.
Confusing the notations for inverse and reciprocal trigonometric functions.
Connecting This to Other Skills
This is a direct application of Inverse Functions (2.13) to the specific case of trigonometric functions.
Solving Trigonometric Equations (2.16) involves using the inverse functions to find principal values.
When Differentiating Trig (4.12) we will see how to differentiate the inverse functions.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?