## What is Tan Function?

Tangent Angle Formula is normally used to calculate the angle of the right-angle triangle. In any given right triangle, the tangent of an angle is the length of the opposite side divided by the length of the adjacent side.

The tangent function, along with sine and cosine, is known to be one of the three most common trigonometric functions. In any given right angle, the tangent of an angle is the length of the opposite side (O) divided by the length of the adjacent side (A).

In a formula, we can simply write it as ‘tan’.

Now tanθ = O/ A

Where,

O = Opposite side

A = Adjacent side

### Tan2x Formula

Trigonometric Formulas like Sin 2x, Cos 2x, Tan 2x are known as double angle formulas because these formulas have double angles in their trigonometric functions.

Let’s discuss Tan2x Formula -

Let’s know how to derive the double angle tan2x formula.

### Derivation of Tan2x Formula

Tan2x Formula = \[\frac{2\text{tan x}}{1 - tan^{2}x}\]

Now, we need to recall the addition formula,

\[tan(a + b) = \frac{\text{tan a + tan b}}{1 - \text{ tan a. tan b}}\]

So, for the double angle formula, here let the values of both the angles be a (a = b)

\[tan(a + a) = \frac{\text{tan a + tan a}}{1 - \text{ tan a. tan a}}\]

Tan 2a = \[\frac{2\text{tan a}}{1 - tan^{2}a}\]

### Values of Tan in Different Values

For angles with their terminal arm in Quadrant II, since the sine function is positive and the cosine is negative, the tangent is negative.

For angles with their terminal arm in Quadrant III, since the sine function is negative and the cosine is negative, the tangent is positive.

For angles with their terminal arm in Quadrant IV, since sine function is negative and cosine is positive, tangent is negative.

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### Tan2x Formulas

Tan2x Formula = \[\frac{2\text{tan x}}{1 - tan^{2}x}\]

We know that tan(x) = sin(x)/cos(x)

Then, tan2x formula = sin(2x)/cos(2x)

Tan 2x can also be written in terms of sin x and cos x,

Tan2x Formula in terms of cos x = \[\frac{2 sin(x) cos(x)}{cos^{2}x - sin^{2}x}\]

### Solved Examples

Question 1) Calculate the tangent angle of a right triangle whose adjacent side and opposite sides are 8 cm and 6 cm respectively?

Solution) Given,

Adjacent side (A)= 8 cm

Opposite side (O)= 6 cm

Using the formula of a tangent:

Tan θ = O/A

Tan θ = 6/8

Tan θ = 0.75

Question 2) Calculate the tangent angle of a right triangle whose adjacent sides and opposite sides are 10 cm and 6 cm respectively?

Solution) Given,

Adjacent side (A) = 10 cm

Opposite side (O) = 6 cm

Using the formula of a tangent:

Tan θ = O/A

Tan θ = 6/10

Tan θ = 0.6

Question 1. What is the Trigonometry Formula?

Answer:

**Basic Formulas:** By using a right-angled triangle as a reference, the trigonometric functions or trigonometric identities are derived: sin θ = Opposite Side/Hypotenuse and cos θ = Adjacent Side/Hypotenuse and tan θ = Opposite Side/Adjacent Side.

Question 2. How do you Solve Trigonometric Identities?

Answer: 4 strategies you can use to solve trigonometric identities

Multiply the denominator by a conjugate.

Get a common denominator.

Split up a fraction into two separate fractions.

Rewrite everything in terms of sine and cosine.

Question 3. Who is the Father of Trigonometry?

Answer:

**Hipparchus of Nicaea:** Hipparchus of Nicaea (in 120 BC) was a Greek astronomer, geographer, and mathematician. Nicaea is considered to be the founder of trigonometry but he is most famous for his incidental discovery of the precession of the equinoxes.