**Here are some common formulas used in trigonometry:**

→ **Sine**: sin(θ) = opposite / hypotenuse

→ **Cosine**: cos(θ) = adjacent / hypotenuse

**→ Tangent**: tan(θ) = opposite / adjacent

**→ Cotangent**: cot(θ) = adjacent / opposite

**→ Secant**: sec(θ) = hypotenuse / adjacent

**→ Cosecant**: csc(θ) = hypotenuse / opposite

**→ Pythagorean identities**:

sin^2(θ) + cos^2(θ) = 1

tan^2(θ) + 1 = sec^2(θ)

cot^2(θ) + 1 = csc^2(θ)

→ **Reciprocal identities**:

csc(θ) = 1 / sin(θ)

sec(θ) = 1 / cos(θ)

cot(θ) = 1 / tan(θ)

**→ Quotient identities**:

tan(θ) = sin(θ) / cos(θ)

cot(θ) = cos(θ) / sin(θ)

sec(θ) = 1 / cos(θ)

csc(θ) = 1 / sin(θ)

**→ Double angle formulas**:

sin(2θ) = 2sin(θ)cos(θ)

cos(2θ) = cos^2(θ) – sin^2(θ) = 1 – 2sin^2(θ) = 2cos^2(θ) – 1

tan(2θ) = (2tan(θ)) / (1 – tan^2(θ))

**→ Half-Angle Formulas**:

sin(θ/2) = ± √[(1 – cos(θ)) / 2]

cos(θ/2) = ± √[(1 + cos(θ)) / 2]

tan(θ/2) = ± √[(1 – cos(θ)) / (1 + cos(θ))]

***Please note that in above formulas ‘θ’ is any angle in radians and not in degree. Also, sometimes you’ll find different notation, like instead of θ it’s been replaced by x, like sinx, cosx, tanx etc.**

#### Inverse Trigonometry

**➫Inverse Sine (arcsine)**: sin^-1(x) = θ, where -π/2 ≤ θ ≤ π/2 and -1 ≤ x ≤ 1

**➫Inverse Cosine (arccosine)**: cos^-1(x) = θ, where 0 ≤ θ ≤ π and -1 ≤ x ≤ 1

**➫Inverse Tangent (arctangent)**: tan^-1(x) = θ, where -π/2 ≤ θ ≤ π/2 and -∞ < x < ∞

**➫Inverse Cotangent (arccotangent)**: cot^-1(x) = θ, where 0 ≤ θ < π and x > -∞, x ≠ 0

**➫Inverse Secant (arcsecant)**: sec^-1(x) = θ, where 0 < θ ≤ π and x > 0, x ≠ 1

**➫Inverse Cosecant (arccosecant)**: csc^-1(x) = θ, where -π/2 < θ < π/2 and x > 0, x ≠ 1

***Please note, the above formulas are provided in radians, and the notations are slightly different from the common notation, like instead of arcsine it’s sin^-1, instead of arccosine it’s cos^-1, etc.**

#### Basics of Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles, particularly right triangles. The word “trigonometry” comes from the Greek words “tri-” (meaning “three”), “gon” (meaning “angle”), and “metron” (meaning “measure”).

Trigonometry is used in a wide range of fields, including mathematics, physics, engineering, and computer science. Some of the key concepts in trigonometry include:

☞ **Trigonometric functions:** Trigonometry defines six main functions — sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc) — which take an angle as an input and return a value between -1 and 1. These functions are used to describe the ratios of the sides of a right triangle, with respect to its angles.

☞ **Inverse trigonometric functions:** The inverse of a trigonometric function is its reciprocal function. For example, the inverse of sine is arcsine, the inverse of cosine is arccosine, and the inverse of tangent is arctangent. These functions take a value between -1 and 1 as an input and return an angle.

☞ **Identities:** Trigonometry includes a wide range of identities that are used to simplify expressions and solve problems. These include the Pythagorean identities, reciprocal identities, quotient identities, double angle identities, and half-angle identities.

☞ **Trigonometric ratios:** Trigonometry helps to calculate the ratio of the sides of a right triangle with respect to its angles, such as Sin(θ) = opposite / hypotenuse, Cos(θ) = adjacent / hypotenuse, and Tan(θ) = opposite / adjacent, etc.

☞ **Degree and radians:** Angle can be measured in two ways, in degree and in radian. Trigonometry uses radians as the standard unit of measurement, but it can also use degree and other units of measurement as well.

☞ **Applications:** Trigonometry has a wide range of applications, including navigation, surveying, physics, engineering, computer graphics, and many more fields.