Tan 2x: Formulas, Applications, and Examples

The tan2x formula for the tangent of a double angle, often denoted as tan⁡(2x)\tan(2x)tan(2x), relates the tangent of an angle to the tangent of twice that angle xxx. Here’s how it’s derived:

Given the double-angle identity for tangent:

tan⁡(2x)=2tan⁡(x)1−tan⁡2(x)\tan(2x) = \frac{2\tan(x)}{1 – \tan^2(x)}tan(2x)=1−tan2(x)2tan(x)​

This formula allows you to express tan⁡(2x)\tan(2x)tan(2x) in terms of tan⁡(x)\tan(x)tan(x), which can be useful in various trigonometric calculations and identities. We get it from basic trigonometry identities, specifically those involving sine and cosine expressed in terms of tangent.

What is Tan2x Formula

The function tan 2x helps in geometry for calculating angles and side lengths in triangles. It’s a trigonometric formula that relates to the tangent of an angle twice as large as x. Trigonometry is a branch of math that focuses on relationships between angles and sides in triangles. It uses ratios like sine, cosine, and tangent (abbreviated as tan) to solve for unknown dimensions in right triangles.

Understanding Tan 2x in Trigonometry

In trigonometry, the formula for tan 2x helps solve different trigonometric problems. It’s a double-angle formula that uses tan x, or sin 2x divided by cos 2x. Since cotangent (cot x) is tan x’s inverse, tan 2x can also be shown as 1/cot 2x.

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Understanding Tan 2x in Trigonometry

Definition of Tan 2x: Tan 2x is a trigonometric function that represents the tangent of twice an angle x. It can be written as:
tan⁡2x=sin⁡2xcos⁡2x\tan 2x = \frac{\sin 2x}{\cos 2x}tan2x=cos2xsin2x​

Double Angle Formulas: To understand tan 2x, we use the double angle identities:

• sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2 \sin x \cos xsin2x=2sinxcosx
• cos⁡2x=cos⁡2x−sin⁡2x\cos 2x = \cos^2 x – \sin^2 xcos2x=cos2x−sin2x

Deriving Tan 2x Using Tangent Addition Formula:

• The tangent addition formula states: tan⁡(a+b)=tan⁡a+tan⁡b1−tan⁡atan⁡b\tan(a + b) = \frac{\tan a + \tan b}{1 – \tan a \tan b}tan(a+b)=1−tanatanbtana+tanb​.
• Apply this to a=b=xa = b = xa=b=x: tan⁡2x=tan⁡(x+x)=tan⁡x+tan⁡x1−tan⁡x⋅tan⁡x=2tan⁡x1−tan⁡2x\tan 2x = \tan(x + x) = \frac{\tan x + \tan x}{1 – \tan x \cdot \tan x} = \frac{2 \tan x}{1 – \tan^2 x}tan2x=tan(x+x)=1−tanx⋅tanxtanx+tanx​=1−tan2x2tanx​

Simplifying Using Double Angle Identities:

• Substitute sin⁡2x\sin 2xsin2x and cos⁡2x\cos 2xcos2x from the double angle formulas into sin⁡2xcos⁡2x\frac{\sin 2x}{\cos 2x}cos2xsin2x​: tan⁡2x=2sin⁡xcos⁡xcos⁡2x−sin⁡2x\tan 2x = \frac{2 \sin x \cos x}{\cos^2 x – \sin^2 x}tan2x=cos2x−sin2x2sinxcosx​

Graphical Interpretation:

• The graph of tan(2x) repeats every π/2, and the graph of tan(x) repeats every π.
• It intersects the x-axis where 2x=nπ2x = n\pi2x=nπ, for integer values of nnn.

Important Notes on Tan 2x Formula

1. Definition: Tan 2x represents the tangent of twice an angle x and can be expressed as:
   tan⁡2x=2tan⁡x1−tan⁡2x\tan 2x = \frac{2 \tan x}{1 – \tan^2 x}tan2x=1−tan2x2tanx​
2. Alternative Representation: Tan 2x can also be written as a ratio of sine 2x to cosine 2x:
   tan⁡2x=sin⁡2xcos⁡2x\tan 2x = \frac{\sin 2x}{\cos 2x}tan2x=cos2xsin2x​
3. Using Trigonometric Identities: Another form of tan 2x involves trigonometric identities:
   tan⁡2x=21−cos⁡2xcos⁡x2cos⁡2x−1\tan 2x = \frac{2 \sqrt{1 – \cos^2 x} \cos x}{2 \cos^2 x – 1}tan2x=2cos2x−121−cos2x​cosx​
4. Further Identity: It can also be expressed using sine and cosine:
   tan⁡2x=2sin⁡x1−2sin⁡2×1−sin⁡2x\tan 2x = \frac{2 \sin x}{1 – 2 \sin^2 x} \sqrt{1 – \sin^2 x}tan2x=1−2sin2x2sinx​1−sin2x​
5. Derivative: The derivative of tan 2x concerning x is:
   ddx(tan⁡2x)=2sec⁡2(2x)\frac{d}{dx} (\tan 2x) = 2 \sec^2 (2x)dxd​(tan2x)=2sec2(2x)
6. Integral: The integral of tan 2x concerning x is:
   ∫tan⁡2x dx=12ln⁡∣sec⁡2x∣+C\int \tan 2x \, dx = \frac{1}{2} \ln |\sec 2x| + C∫tan2xdx=21​ln∣sec2x∣+C

These notes highlight various forms and applications of the tan 2x formula in calculus and trigonometry, useful for solving equations and integrating functions involving double angles.

Example 1: Finding tan 2x given tan x

Problem: If tan⁡x=34\tan x = \frac{3}{4}tanx=43​, find tan⁡2x\tan 2xtan2x.

Solution:

1. Use the formula: tan⁡2x=2tan⁡x1−tan⁡2x\tan 2x = \frac{2 \tan x}{1 – \tan^2 x}tan2x=1−tan2x2tanx​.
2. Substitute tan⁡x=34\tan x = \frac{3}{4}tanx=43​:
   tan⁡2x=2⋅341−(34)2\tan 2x = \frac{2 \cdot \frac{3}{4}}{1 – \left(\frac{3}{4}\right)^2}tan2x=1−(43​)22⋅43​​
3. Calculate the numerator and denominator:
   tan⁡2x=641−916=6416−916=64716\tan 2x = \frac{\frac{6}{4}}{1 – \frac{9}{16}} = \frac{\frac{6}{4}}{\frac{16 – 9}{16}} = \frac{\frac{6}{4}}{\frac{7}{16}}tan2x=1−169​46​​=1616−9​46​​=167​46​​
4. Simplify:
   tan⁡2x=64⋅167=247\tan 2x = \frac{6}{4} \cdot \frac{16}{7} = \frac{24}{7}tan2x=46​⋅716​=724​

Answer: tan⁡2x=247\tan 2x = \frac{24}{7}tan2x=724​.

Example 2: Derivative of tan 2x

Problem: Find ddx(tan⁡2x)\frac{d}{dx} (\tan 2x)dxd​(tan2x).

Solution:

1. Use the derivative formula: ddx(tan⁡2x)=2sec⁡2(2x)\frac{d}{dx} (\tan 2x) = 2 \sec^2 (2x)dxd​(tan2x)=2sec2(2x).
2. Apply the formula:
   ddx(tan⁡2x)=2sec⁡2(2x)\frac{d}{dx} (\tan 2x) = 2 \sec^2 (2x)dxd​(tan2x)=2sec2(2x)

Answer: ddx(tan⁡2x)=2sec⁡2(2x)\frac{d}{dx} (\tan 2x) = 2 \sec^2 (2x)dxd​(tan2x)=2sec2(2x).

Example 3: Integral of tan 2x

Problem: Evaluate ∫tan⁡2x dx\int \tan 2x \, dx∫tan2xdx.

Solution:

1. Use the integral formula: ∫tan⁡2x dx=12ln⁡∣sec⁡2x∣+C\int \tan 2x \, dx = \frac{1}{2} \ln |\sec 2x| + C∫tan2xdx=21​ln∣sec2x∣+C.
2. Apply the formula:
   ∫tan⁡2x dx=12ln⁡∣sec⁡2x∣+C\int \tan 2x \, dx = \frac{1}{2} \ln |\sec 2x| + C∫tan2xdx=21​ln∣sec2x∣+C

Answer: ∫tan⁡2x dx=12ln⁡∣sec⁡2x∣+C\int \tan 2x \, dx = \frac{1}{2} \ln |\sec 2x| + C∫tan2xdx=21​ln∣sec2x∣+C.