Trig Identities.md

#z_Legacy/Maths/A-Levels/Pure/Trig-Identities

$\cos^2x + \sin^2x \equiv 1$

[!example]- Proof

• Consider a right triangle with an angle $x$.
• Let the hypotenuse be of length 1.
• According to the definitions of sine and cosine, we have: $$\sin x = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{1} = a$$ and $$\cos x = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{1} = b,$$
• By the Pythagorean theorem, we know that in a right triangle: $$a^2 + b^2 = 1^2$$ Substituting the values of $a$ and $b$ in terms of $\sin x$ and $\cos x$, we get: $$\sin ^2(x) + \cos^2(x) = 1$$

$1 + \tan^2(x) \equiv \sec^2(x)$

[!example]- Proof

• Take $$\sin ^2(x) + \cos^2(x) = 1$$
• Divide by $sin^2(x)$ to get $$\frac{sin ^2(x)}{sin ^2(x)} + \frac{cos^2(x)}{sin ^2(x)} = \frac{1}{sin ^2(x)}$$
• Rearrange to get $$1 + \left (\frac{cos(x)}{sin(x)} \right)^2 \equiv \left (\frac{1}{sin(x)} \right)^2$$
• Simplify to get $$1 + tan^2(x) \equiv sec^2(x)$$

$a\sin(x) \pm b\cos(x)\equiv R\sin(x \pm \alpha), R\cos(x \pm \alpha)$

[!example]- Equation $$sign(a)\left(\sqrt{a^{2}+b^{2}}\right)\sin\left(x + \arctan\left(\frac{b}{a}\right)\right)$$ or $$sign(b)\left(\sqrt{a^{2}+b^{2}}\right)\cos\left(x-\arctan\left(\frac{a}{b}\right)\right)$$