### Trigonometric Functions

**Trigonometric Functions for JEE and CBSE board exams**

**Trigonometric ratios**

- Sine (sin θ) = Length of opposite side / Hypotenuse
- Cosine (cos θ) = Length of adjacent side / Hypotenuse
- Tangent (tan θ) = Opposite side / Adjacent side

**Trigonometric identities**

- Pythagorean identity: sin^2 θ + cos^2 θ = 1
- Sum and difference formulas:
- sin(A ± B) = sin A cos B ± cos A sin B
- cos(A ± B) = cos A cos B ∓ sin A sin B
- tan(A ± B) = (tan A ± tan B) / (1∓ tan A tan B)
- Product-to-sum formulas:
- sin A cos B = ½ (sin(A + B) + sin(A - B))
- cos A sin B = ½ (sin(A + B) - sin(A - B))

**Graphs of trigonometric functions**

**Sine:**A periodic wave that oscillates between -1 and 1.**Cosine:**A periodic wave that oscillates between -1 and 1, shifted by π/2 radians from the sine curve.**Tangent:**A non-periodic wave that increases or decreases without bound as θ approaches π/2 or -π/2.

**Inverse trigonometric functions**

**Arcsine (sin^-1 θ):**The angle whose sine is θ.**Arccosine (cos^-1 θ):**The angle whose cosine is θ.**Arctangent (tan^-1 θ):**The angle whose tangent is θ.

**Applications of trigonometry**

- Finding the height of a building using trigonometric ratios.
- Determining the distance to a star by measuring the angle of parallax.
- Calculating the angle of elevation/depression of an object.