Trigonometry

Trigonometry originates from the study of triangles, particularly, the right angled ones.

Pythagoras Theorem

Pythagoras states that the expression is always be true for any three sides of a right angle triangle, where c is the hypothenuse.

Trigonometry

Trigonometric Identities

We obtained the second expression by dividing the first expression throughout by ${\sin }^2(x)$ and the third by dividing the first expression by ${\cos }^2(x)$

Sine rule

Cos rule

Physics related concept

let $\sin \omega t$ have a wavelength of $\lambda$

The relationship between $\omega$ and $\lambda$ is.

thus

Double angle formula

Harmonic Addition Theorem

Express $R\sin (\theta +a)$ as $A\cos \theta +B\sin \theta$ .

Thus

Reverse Factor Formulae

Additional Insights (Very useful)

Let us see the properties of these trigo identities.
This section aims to help us find a shortcut to solving questions like: Use Compound angle formula to prove that $\cos \frac{19\pi }{12}=\frac{\sqrt{6}-\sqrt{2}}{4}$ etc...

let us define $n$ as

where $R$ is the remainder. And $k,a,R,n\in \mathbb{Z}$.
Look at Proof for Symmetries in Trigo Identities. We thus get.

Joshua's approximation method. Estimating $cos$ $sin$ $tan$ without using their respective functions on a calculator

Suppose we want to find the value of

Using the approximation that $\pi \approx \frac{22}{7}$. And reducing it using the Proof for Symmetries in Trigo Identities formulas

Next, having reduced $\theta <\frac{\pi }{2}$. using the Maclaurin Series expansion for $\cos$ up to the $x^4$ term.

We see that

and using a calculator

Comparing the two solution, my method is accurate up to 2dp with an error of $0.001$ :)