Special Angles and Trigonometric Identities

Name: Special Angles and Trigonometric Identities
Author: StemPy Lab

The exact ratios of 0°, 30°, 45°, 60° and 90° from two simple triangles, and the identities that link the ratios, used to simplify and prove.

3 lessons, about 1 h 5 min3 simulations11 problems3 practice questionsPython in every lesson

Course outline (3 lessons)

Lesson 1about 25 min
Ratios of special angles
Two simple triangles give the exact ratios of $30^\circ$ , $45^\circ$ and $60^\circ$ : half of a square, and half of an equilateral triangle. As the angle shrinks to $0^\circ$ or grows to $90^\circ$ , the ratios reach $0$ and $1$ , and $\tan 90^\circ$ is not defined. With the table we can work out expressions and find an angle from its ratio.
1 simulation1 practice question1 Python program4 problems1 Python lab
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Lesson 2about 20 min
The three Pythagorean identities
In a right triangle, Pythagoras gives $\sin^2 A + \cos^2 A = 1$ for every acute angle $A$ . Dividing by $\cos^2 A$ gives $1 + \tan^2 A = \sec^2 A$ , and dividing by $\sin^2 A$ gives $1 + \cot^2 A = \text{cosec}^2 A$ . With these, every ratio can be written in terms of any one of them. Angles that add to $90^\circ$ swap their ratios.
1 simulation1 practice question1 Python program3 problems
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Lesson 3about 20 min
Proving identities
To prove an identity, start from one side, usually the more complicated one, and change it step by step until it becomes the other side. Write everything in sines and cosines when stuck, use the three identities, and never treat the identity as known while proving it. Checking a few angles can show that a claim is false, but only a proof shows that it is always true.
1 simulation1 practice question1 Python program4 problems1 Python lab1 question set1 note
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10 questions · 20 min

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