Problem 100 - Precalculus

Question

Explain how to determine the double-angle formula for \(\tan (2 x)\) using the double-angle formulas for \(\cos (2 x)\) and \(\sin (2 x)\)

Step-by-Step Solution

Answer

The double-angle formula for \( \tan(2x) \) is \( \frac{2 \tan(x)}{1 - \tan^2(x)} \).

Step 1: Recall Double-Angle Formulas

The double-angle formulas for sine and cosine are essential to derive the formula for tangent. Recall that \( \sin(2x) = 2 \sin(x) \cos(x) \) and \( \cos(2x) = \cos^2(x) - \sin^2(x) \). These will be used to find \( \tan(2x) \).

Step 2: Express \( \tan(2x) \) as \( \frac{\sin(2x)}{\cos(2x)} \)

Tangent of an angle is defined as the ratio of sine to cosine. Therefore, \( \tan(2x) = \frac{\sin(2x)}{\cos(2x)} \). Use the double-angle formulas to replace \( \sin(2x) \) and \( \cos(2x) \) in this expression.

Step 3: Substitute Double-Angle Expressions

Substitute the expressions from Step 1 into the equation: \( \tan(2x) = \frac{2 \sin(x) \cos(x)}{\cos^2(x) - \sin^2(x)} \). This forms a fraction that represents \( \tan(2x) \) in terms of \( \sin(x) \) and \( \cos(x) \).

Step 4: Simplify the Expression

To simplify, divide both the numerator and denominator by \( \cos^2(x) \): \( \tan(2x) = \frac{2 \frac{\sin(x)}{\cos(x)}}{1 - \frac{\sin^2(x)}{\cos^2(x)}} \). This becomes \( \tan(2x) = \frac{2 \tan(x)}{1 - \tan^2(x)} \) by substituting \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). This is the double-angle formula for tangent.

Key Concepts

Tangent

The tangent of an angle in trigonometry is calculated as the ratio of the sine of that angle to its cosine. The formula for tangent is given by

• \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).

For a double angle, this concept extends to

• \( \tan(2x) = \frac{\sin(2x)}{\cos(2x)} \).

By using the double-angle formulas, this relationship can be rewritten in terms of the tangent of \( x \) itself, leading us to the expression

• \( \tan(2x) = \frac{2 \tan(x)}{1 - \tan^2(x)} \).

This expression helps in simplifying computations and is particularly helpful in solving trigonometric equations and analyzing periodic functions.

Sine

Sine is an elementary function in trigonometry describing a ratio of sides in a right triangle. Unlike tangent, sine does not need to involve both the adjacent and opposite sides. Instead, it is simply the ratio of the opposite side to the hypotenuse

• \( \sin(x) = \frac{\text{opposite}}{\text{hypotenuse}} \).

For a double angle, we have a specific formula that helps manipulate expressions and simplify complex trigonometric equations. The double-angle formula for sine is

• \( \sin(2x) = 2 \sin(x) \cos(x) \).

This formula is derived from the sum identities for sine, where

• \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \).

Using this identity to handle expressions involving double angles can simplify various trigonometric tasks.

Cosine

Cosine, another cornerstone of trigonometry, is the ratio of the adjacent side to the hypotenuse of a right triangle

• \( \cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}} \).

When analyzing double angles such as \( 2x \), the double-angle formula for cosine is indispensable. This is written as

• \( \cos(2x) = \cos^2(x) - \sin^2(x) \).

Alternatively, this can be expressed using only cosine or only sine by employing identities like

• \( \cos(2x) = 2 \cos^2(x) - 1 \) or \( \cos(2x) = 1 - 2 \sin^2(x) \).

This versatility allows for multiple ways to express and work with trigonometric problems, making cosine double-angle formulas very handy for various derivations and simplifications.

Trigonometric Identities

Trigonometric identities are fundamental expressions connecting the primary trigonometric functions. These include sine, cosine, and tangent. They are useful tools for simplifying and solving equations. In particular, the identities used in the double-angle formula for tangent are crucial for various applications in mathematics and engineering.The most useful identities are

• Pythagorean identities: such as \( \sin^2(x) + \cos^2(x) = 1 \).
• Sum and difference formulas like \( \sin(a\pm b) \) and \( \cos(a\pm b) \).
• Even-odd identities: \( \sin(-x) = -\sin(x) \) and \( \cos(-x) = \cos(x) \).

Applying these identities allows us to derive expressions like the double-angle formulas, which are key to breaking down and understanding more complex trigonometric functions. Mastery of these identities simplifies the computation of angles and assists in the transformation of trigonometric expressions.