P3 Trigonometry: Identities and Harmonic Form

Unlocking Trig's Secret Moves 🤸

Introduction

1. Introduction

\sin(A+B)

sec(\theta) = \frac{1}{cos(\theta)}

sin^2(\theta) + cos^2(\theta) = 1

Worked example

Let's Crack This Trig Puzzle 🚀

3tan^2(x) = 5sec(x) - 1

1
$tan$
$3(sec^2(x) - 1) = 5sec(x) - 1$
2
$sec(x)$
$3sec^2(x) - 3 = 5sec(x) - 1 \\ 3sec^2(x) - 5sec(x) - 2 = 0$
3
$3 \times -2 = -6$
$(3sec(x) + 1)(sec(x) - 2) = 0$
4
$sec(x)$
$sec(x) = -\frac{1}{3} \quad \text{or} \quad sec(x) = 2$
5
$sec(x)$
$cos(x) = -3 \quad (\text{no solution}) \\ cos(x) = \frac{1}{2} \quad (\text{valid})$
6
$cos(x) = \frac{1}{2}$
$x = cos^{-1}(0.5) = 60^\circ \\ x = 360^\circ - 60^\circ = 300^\circ \\ \text{Final Answers: } x = 60.0^\circ, 300.0^\circ$

Answer

x = cos^{-1}(0.5) = 60^\circ \\ x = 360^\circ - 60^\circ = 300^\circ \\ \text{Final Answers: } x = 60.0^\circ, 300.0^\circ

\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B

\sin(A+B)

Worked example

Cracking the Code for sin(75°) 🕵️‍♀️

\sin(75°)

y = a\sin\theta

R

Worked example

Let's Beat This Boss Level 🚀

3\cos\theta + 4\sin\theta

1
$R\cos(\theta - \alpha)$
$3\cos\theta + 4\sin\theta \equiv R\cos(\theta - \alpha) \equiv R(\cos\theta\cos\alpha + \sin\theta\sin\alpha)$
2
$\cos\theta$
$R\cos\alpha = 3 \quad \text{and} \quad R\sin\alpha = 4$
3
$R$
$R^2 = 3^2 + 4^2 = 9 + 16 = 25 \implies R = 5$
4
$\alpha$
$\frac{R\sin\alpha}{R\cos\alpha} = \tan\alpha = \frac{4}{3} \implies \alpha = \arctan\left(\frac{4}{3}\right) = 53.1^\circ$
5
$Now we can rewrite the original equation using our R-form. This looks much simpler to solve!$
$5\cos(\theta - 53.1^\circ) = 2.5 \implies \cos(\theta - 53.1^\circ) = \frac{2.5}{5} = 0.5$
6
$(\theta - 53.1^\circ)$
$\theta - 53.1^\circ = \arccos(0.5) = 60^\circ, \; 360^\circ - 60^\circ, \; ... \\ \theta - 53.1^\circ = 60^\circ \implies \theta = 113.1^\circ \\ \theta - 53.1^\circ = 300^\circ \implies \theta = 353.1^\circ$

Answer

\theta - 53.1^\circ = \arccos(0.5) = 60^\circ, \; 360^\circ - 60^\circ, \; ... \\ \theta - 53.1^\circ = 60^\circ \implies \theta = 113.1^\circ \\ \theta - 53.1^\circ = 300^\circ \implies \theta = 353.1^\circ

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