Find All Solutions 2tan 2x 1
Tan 2x Formula
Tan 2x is an important trigonometric function. Tan 2x formula is one of the very commonly used double angle trigonometric formulas and can be expressed in terms of different trigonometric functions such as tan x, cos x, and sin x. As we know that tan x is the ratio of sine and cosine function, therefore the tan 2x identity can also be expressed as the ratio of sin 2x and cos 2x.
In this article, we will learn the tan 2x formula, its proof and express it in terms of different trigonometric functions. We will also determine the derivative and integral of tan 2x.
| 1. | What is Tan 2x in Trigonometry? |
| 2. | Tan 2x Formula |
| 3. | Tan 2x Formula Proof |
| 4. | Graph of Tan 2x |
| 5. | Tan 2x Formula in Terms of Cos |
| 6. | Derivative and Integral of Tan 2x |
| 7. | FAQs on Tan 2x Formula |
What is Tan 2x in Trigonometry?
Tan 2x is a trigonometric function and has a formula that is used to solve various problems in trigonometry. Tan 2x is an important double angle formula, that is, a trigonometry formula where the angle is doubled. It can be expressed in terms of tan x and also as a ratio of sin 2x and cos 2x. Let us see the tan 2x formula:
Tan 2x Formula
The formula for tan 2x identity is given as:
- tan 2x = 2tan x / (1−tan2x)
- tan 2x = sin 2x/cos 2x
Tan 2x Formula Proof
Tan 2x formula can be derived using two different methods. First, we will use the angle addition formula for the tangent function to derive the tan 2x identity. Note that we can write the double angle 2x as 2x = x + x. We will use the following trigonometric formula to prove the formula for tan 2x:
- tan (a + b) = (tan a + tan b)/(1 - tan a tan b)
We have
tan 2x = tan (x + x)
= (tan x + tan x)/(1 - tan x tan x)
= 2 tan x/(1 - tan2x)
Hence, we have derived the tan 2x formula using the angle sum formula of the tangent function.
Now, we will derive the tan 2x formula by expressing tan as a ratio of sin and cos. We will use the following trigonometric formulas:
- tan x = sin x/ cos x
- sin 2x = 2 sin x cos x
- cos 2x = cos2x - sin2x
Using the above formulas, we have
tan 2x = sin 2x/cos 2x
= 2 sin x cos x/(cos2x - sin2x)
Divide the numerator and denominator of 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 we have derived the tan 2x formula by expressing it as a ratio of sin 2x and cos 2x.
Graph of Tan 2x
The graph of tan 2x looks similar to the graph of tan x. We know that the period of tan x is π. Since the period of tan bx is given by π/|b|, the period of tan 2x is π/2. Given below is the graph of tan 2x and as we can observe from the graph, the value of tan 2x repeats after every π/2 radians.
Tan 2x in Terms of Cos
We can derive the tan 2x formula in terms of cos. We will use the following trigonometric formulas to express tan 2x in terms of cos x.
- tan x = sin x/ cos x
- sin 2x = 2 sin x cos x
- cos 2x = 2 cos2x - 1
- sin x = √(1 - cos2x)
Using the above formulas, we have
tan 2x = sin 2x/ cos 2x
= 2 sin x cos x/(2 cos2x - 1)
= [2 cos x/(2 cos2x - 1)]√(1 - cos2x)
Similarly, we can write tan 2x in terms of sin using the trigonometric identities.
tan 2x = [2 sin x/(1 - 2 sin2x)]√(1 - sin2x)
Derivative and Integral of Tan 2x
We know that the derivative of trigonometric function tan x is given by sec2x. The derivative of tan 2x can be calculated using different methods such as the chain rule and quotient rule. Let us determine the derivative of tan 2x using the chain rule.
d(tan 2x)/dx = d(tan 2x)/d(2x) × d(2x)/dx
= sec22x × 2
= 2 sec2(2x)
Hence the derivative of tan 2x is 2 sec2(2x).
Now, we will determine the integral of tan 2x. We know that the integral of tan x is -ln |cos x| + C or ln |sec x| + C. Using the formulas of integration, the integral of tan 2x is given by,
∫ tan 2x = (-1/2) ln |cos 2x| + C
= (1/2) ln |sec 2x| + C
Hence the integral of tan 2x is given by (-1/2) ln |cos 2x| + C or (1/2) ln |sec 2x| + C.
Important Notes on Tan 2x Formula
- tan 2x = 2tan x / (1 − tan2x)
- tan 2x = sin 2x/cos 2x
- The derivative of tan 2x is 2 sec2(2x)
- The integral of tan 2x is (-1/2) ln |cos 2x| + C or (1/2) ln |sec 2x| + C.
Topics Related to Tan 2x
- Tangent Formulas
- Cos 2x
Tan 2x Examples
-
Example 1: Determine the value of tan 2x if tan x = 4/3.
Solution: We know that tan 2x = 2tan x / (1 − tan2x). If tan x = 4/3, we have tan2x = 16/9.
tan 2x = 2tan x / (1 − tan2x)
= 2(4/3)/(1 - 16/9)
= (8/3)/(-7/9)
= -24/7
Answer:tan 2x = -24/7
-
Example 2: Find the value of tan 2x if sin x = 12/13 and cos x = 5/13
Solution: We know that tan 2x = sin 2x/ cos 2x
= 2 sin x cos x /(cos2x - sin2x)
= [2 × (12/13) × (5/13)] / [(25/169) - (144/169)]
= (120/169) / (-119/169)
= -120/119
Answer: tan 2x = -120/119
go to slidego to slide
Breakdown tough concepts through simple visuals.
Math will no longer be a tough subject, especially when you understand the concepts through visualizations.
Book a Free Trial Class
Practice Questions on Tan 2x
go to slidego to slide
FAQs on Tan 2x Formula
What is Tan 2x in Trigonometry?
Tan 2x formula is one of the very commonly used double angle trigonometric formulas and can be expressed in terms of different trigonometric functions such as tan x, cos x, and sin x.
What is Tan 2x Formula?
Tan 2x formula can be expressed in different ways such as:
- tan 2x = 2tan x / (1 − tan2x)
- tan 2x = sin 2x/cos 2x
- tan 2x = [2 cos x/(2 cos2x - 1)]√(1 - cos2x)
- tan 2x = [2 sin x/(1 - 2 sin2x)]√(1 - sin2x)
What is the Domain and Range of Tan 2x?
The domain of tan 2x consists of all real numbers except those where tan 2x is not defined. So, the domain of tan 2x is R - {(2n + 1)π/4, n ∈ Z}. The range of tan 2x is all real numbers, that is, R.
How to Find the Derivative of Tan 2x?
The derivative of tan 2x can be calculated using different methods such as the chain rule and quotient rule. The derivative of tan 2x is 2 sec2(2x).
What is the Integral of Tan 2x?
The integral of tan 2x is given by (-1/2) ln |cos 2x| + C or (1/2) ln |sec 2x| + C.
What is the Formula of Tan 2x in Terms of Cos x and Sin x?
Tan 2x formula in terms of sin x and cos x is given by tan 2x = 2 sin x cos x /(cos2x - sin2x)
Source: https://www.cuemath.com/tan2x-formula/