Tan 2x Formula

Introduction 

Tan 2x is a double-angle trigonometric function that can be represented as tan x, sin x, and cos x. The tan angle formula is used to calculate the angle of a right triangle. It has been found that the tangent to the angle of a right triangle is equal to the length of the opposite side divided by the length of the adjacent side. It can be expressed as tan 2x or as a ratio of sin 2x to cos 2x. Since the reciprocal of tan x is cot x, we can write tan 2x as the reciprocal of cot 2x, that is, tan 2x = 1/cot 2x. 

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 tan θ = O/ A

Where,

O = Opposite side

A = Adjacent side

Tan 2x Formula

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

The tan 2x expression can be expressed in two different formats. This can only be expressed as a tangent function and as a combination of sine and cosine functions. The formula for the tan 2x identity is: 

Tan 2x= \[\frac{sin2x}{cos2x}\]

Tan 2x=\[\frac{2 tanx }{1-tan^{2}x}\]

Let’s know how to derive the double angle tan 2x formula.

Tan 2x Formula Derivation 

Tan 2x expressions can be derived by using two different methods. First, we derive tan 2x identity using the angle addition formula of the tangent function. Note that you can write a double angle 2x as 2x = x + x. Prove the expression for tan 2x using the following trigonometric expression. 

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

tan 2x = tan (x + x)=\[\frac{tan x+tan x}{1-tan x . tan x}\]

=\[\frac{2 tanx }{1-tan^{2}x}\]

Therefore, we have derived the tan 2x formula using the angle sum formula of the tangent function.  

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, the tangent is negative.

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Tan 2x Identity Proof in Form of sinx and cosx 

Derive the tan 2x formula by expressing tan as a ratio of sin and cos by using the following trigonometric formulas:

tan x = sin x/cos x

sin 2x = 2 sin x . cos x

cos 2x = cos ²x – sin²x

Using the formulas given above, we get

tan 2x = sin 2x/cos 2x

= (2 sin x.cos x)/(cos²x–sin²x)

Dividing both the numerator and denominator of the equation 2 sin x . cos x/(1 –2 sin²x) by cos²x

tan 2x = [2 sin x. cos x/cos²x]/[(cos²x–sin²x)/cos²x]

= (2 sin x/cos x)/[1– (sin²x/cos²x)]

= 2 tan x/(1 –tan²x)

Hence, the tan 2x formula is derived and expressed as a ratio of sin 2x and cos 2x.

List of all Tan 2x formulas:

1. tan(2x)= 2tanx/1–tanx.tanx

2. tan 2x = sin 2x/cos 2x 

3. Derivative of tan 2x is 2 sec²(2x)

4. Integral of tan2x is (–1/2) ln |cos 2x| + C or (1/2) ln |sec 2x| + C.

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 3) If tan x = 3/4, find the value of tan 2x. 

Solution  I know the formula tan 2x = 2.tan x / (1–tan²x). 

Therefore, if tan x = ¾, then  tan²x = 9/16. 

Tan 2x = 2 (3/4) / [1–(9/16)]

= (3/2) / (7/16) 

= 24/7

Question 4) If sin x = 12/13 and cos x = 5/13, determine the value of tan 2x. 

 Solution  tan 2x = [2 × (12/13) × (5/13)] / [(25/169) – (144/169)] 

 = (120/169) /–(119/169) 

 = –120/119 

 Answer: tan 2x = – 120/119