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−tan2(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 helpful 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’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 representing the tangent of twice an angle x. It can be written as:
tan2x=sin2xcos2x\tan 2x = \frac{\sin 2x}{\cos 2x}tan2x=cos2xsin2x
Double Angle Formulas: To understand tan 2x, we use the double angle identities:
- sin2x=2sinxcosx\sin 2x = 2 \sin x \cos xsin2x=2sinxcosx
- cos2x=cos2x−sin2x\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)=tana+tanb1−tanatanb\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: tan2x=tan(x+x)=tanx+tanx1−tanx⋅tanx=2tanx1−tan2x\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 sin2x\sin 2xsin2x and cos2x\cos 2xcos2x from the double angle formulas into sin2xcos2x\frac{\sin 2x}{\cos 2x}cos2xsin2x: tan2x=2sinxcosxcos2x−sin2x\tan 2x = \frac{2 \sin x \cos x}{\cos^2 x – \sin^2 x}tan2x=cos2x−sin2x2sinxcosx
Graphical Interpretation:
- The tan(2x) graph repeats every π/2, and the tan(x) repeats every π.
- It intersects the x-axis where 2x=nπ2x = n\pi2x=nπ, for integer values of nnn.
Essential Notes on Tan 2x Formula
- Definition: Tan 2x represents the tangent of twice an angle x and can be expressed as:
tan2x=2tanx1−tan2x\tan 2x = \frac{2 \tan x}{1 – \tan^2 x}tan2x=1−tan2x2tanx - Alternative Representation: Tan 2x can also be written as a ratio of sine 2x to cosine 2x:
tan2x=sin2xcos2x\tan 2x = \frac{\sin 2x}{\cos 2x}tan2x=cos2xsin2x - Using Trigonometric Identities: Another form of tan 2x involves trigonometric identities:
tan2x=21−cos2xcosx2cos2x−1\tan 2x = \frac{2 \sqrt{1 – \cos^2 x} \cos x}{2 \cos^2 x – 1}tan2x=2cos2x−121−cos2xcosx - Further Identity: It can also be expressed using sine and cosine:
tan2x=2sinx1−2sin2×1−sin2x\tan 2x = \frac{2 \sin x}{1 – 2 \sin^2 x} \sqrt{1 – \sin^2 x}tan2x=1−2sin2x2sinx1−sin2x - Derivative: The derivative of tan 2x concerning x is:
ddx(tan2x)=2sec2(2x)\frac{d}{dx} (\tan 2x) = 2 \sec^2 (2x)dxd(tan2x)=2sec2(2x) - Integral: The integral of tan 2x concerning x is:
∫tan2x dx=12ln∣sec2x∣+C\int \tan 2x \, dx = \frac{1}{2} \ln |\sec 2x| + C∫tan2xdx=21ln∣sec2x∣+C
These notes highlight various forms and applications of the tan 2x formula in calculus and trigonometry, which helps solve equations and integrate double-angle functions.
Example 1: Finding tan 2x given tan x
Problem: If tanx=34\tan x = \frac{3}{4}tanx=43, find tan2x\tan 2xtan2x.
Solution:
- Use the formula: tan2x=2tanx1−tan2x\tan 2x = \frac{2 \tan x}{1 – \tan^2 x}tan2x=1−tan2x2tanx.
- Substitute tanx=34\tan x = \frac{3}{4}tanx=43:
tan2x=2⋅341−(34)2\tan 2x = \frac{2 \cdot \frac{3}{4}}{1 – \left(\frac{3}{4}\right)^2}tan2x=1−(43)22⋅43 - Calculate the numerator and denominator:
tan2x=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=1−16946=1616−946=16746 - Simplify:
tan2x=64⋅167=247\tan 2x = \frac{6}{4} \cdot \frac{16}{7} = \frac{24}{7}tan2x=46⋅716=724
Answer: tan2x=247\tan 2x = \frac{24}{7}tan2x=724.
Example 2: Derivative of tan 2x
Problem: Find ddx(tan2x)\frac{d}{dx} (\tan 2x)dxd(tan2x).
Solution:
- Use the derivative formula: ddx(tan2x)=2sec2(2x)\frac{d}{dx} (\tan 2x) = 2 \sec^2 (2x)dxd(tan2x)=2sec2(2x).
- Apply the formula:
ddx(tan2x)=2sec2(2x)\frac{d}{dx} (\tan 2x) = 2 \sec^2 (2x)dxd(tan2x)=2sec2(2x)
Answer: ddx(tan2x)=2sec2(2x)\frac{d}{dx} (\tan 2x) = 2 \sec^2 (2x)dxd(tan2x)=2sec2(2x).
Example 3: Integral of tan 2x
Problem: Evaluate ∫tan2x dx\int \tan 2x \, dx∫tan2xdx.
Solution:
- Use the integral formula: ∫tan2x dx=12ln∣sec2x∣+C\int \tan 2x \, dx = \frac{1}{2} \ln |\sec 2x| + C∫tan2xdx=21ln∣sec2x∣+C.
- Apply the formula:
∫tan2x dx=12ln∣sec2x∣+C\int \tan 2x \, dx = \frac{1}{2} \ln |\sec 2x| + C∫tan2xdx=21ln∣sec2x∣+C
Answer: ∫tan2x dx=12ln∣sec2x∣+C\int \tan 2x \, dx = \frac{1}{2} \ln |\sec 2x| + C∫tan2xdx=21ln∣sec2x∣+C.
FAQs
The tangent function, tan (x), is a trigonometric function representing the ratio of the opposite side to the adjacent side in a right triangle.
Fundamental properties include its periodicity, where tan (2𝑥) has a period of π, and the formula simplifies complex trigonometric expressions.
It simplifies complex trigonometric equations by expressing tan tan(2x) in terms of tan(x), making it easier to solve.
