List of Trigonometry Formulas

 The Trigonometric formulas or Identities are the equations which are true in the case of Right-Angled Triangles. Some of the special trigonometric identities are given below –

1. Pythagorean Identities

• sin²θ + cos²θ = 1
• tan2θ + 1 = sec2θ
• cot2θ + 1 = cosec2θ
• sin 2θ = 2 sin θ cos θ
• cos 2θ = cos²θ – sin²θ
• tan 2θ = 2 tan θ / (1 – tan²θ)
• cot 2θ = (cot²θ – 1) / 2 cot θ

2. Sum and Difference identities-

For angles u and v, we have the following relationships:

• sin(u + v) = sin(u)cos(v) + cos(u)sin(v)
• cos(u + v) = cos(u)cos(v) – sin(u)sin(v)
• $\begin{array}{l}𝑡𝑎𝑛(𝑢+𝑣)=\frac{𝑡𝑎𝑛(𝑢)+𝑡𝑎𝑛(𝑣)}{1-𝑡𝑎𝑛(𝑢)𝑡𝑎𝑛(𝑣)}\end{array}$
• sin(u – v) = sin(u)cos(v) – cos(u)sin(v)
• cos(u – v) = cos(u)cos(v) + sin(u)sin(v)
• $\begin{array}{l}𝑡𝑎𝑛(𝑢-𝑣)=\frac{𝑡𝑎𝑛(𝑢)-𝑡𝑎𝑛(𝑣)}{1+𝑡𝑎𝑛(𝑢)𝑡𝑎𝑛(𝑣)}\end{array}$

3. If A, B and C are angles and a, b and c are the sides of a triangle, then,

Sine Laws

• a/sinA = b/sinB = c/sinC

Cosine Laws

• c2 = a2 + b2 – 2ab cos C
• a2 = b2 + c2 – 2bc cos A
• b2 = a2 + c2 – 2ac cos B

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Trigonometry Identities

The three important trigonometric identities are:

• sin²θ + cos²θ = 1
• tan²θ + 1 = sec²θ
• cot²θ + 1 = cosec²θ

Euler’s Formula for trigonometry

As per the euler’s formula, 

eix = cos x + i sin x

Where x is the angle and i is the imaginary number. 

$\begin{array}{l}\sin 𝑥=\frac{{𝑒}^{𝑖𝑥}-{𝑒}^{-𝑖𝑥}}{2𝑖}\\ \cos 𝑥=\frac{{𝑒}^{𝑖𝑥}+{𝑒}^{-𝑖𝑥}}{2}\\ \tan 𝑥=\frac{({𝑒}^{𝑖𝑥}-{𝑒}^{-𝑖𝑥})}{𝑖({𝑒}^{𝑖𝑥}+{𝑒}^{-𝑖𝑥})}\end{array}$

Trigonometry Basics

The three basic functions in trigonometry are sine, cosine and tangent. Based on these three functions the other three functions that are cotangent, secant and cosecant are derived. 

All the trigonometrical concepts are based on these functions. Hence, to understand trigonometry further we need to learn these functions and their respective formulas at first.

If θ is the angle in a right-angled triangle, then 

Sin θ = Perpendicular/Hypotenuse

Cos θ = Base/Hypotenuse

Tan θ = Perpendicular/Base

Perpendicular is the side opposite to the angle θ.

The base is the adjacent side to the angle θ.

The hypotenuse is the side opposite to the right angle

The other three functions i.e. cot, sec and cosec depend on tan, cos and sin respectively, such as:

Cot θ = 1/tan θ

Sec θ = 1/cos θ

Cosec θ = 1/sin θ

Hence, 

Cot θ = Base/Perpendicular

Sec θ = Hypotenuse/Base

Cosec θ = Hypotenuse/Perpendicular