Problem 10 - Algebra 2

Question

Use an angle sum identity to verify each identity. $$ \tan 2 \theta=\frac{2 \tan \theta}{1-\tan ^{2} \theta} $$

Step-by-Step Solution

Answer

After simplifying both sides of the equation, it's confirmed that \(\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^{2} \theta}\), verifying the identity.

Step 1: Establish the Formula for Double the Angle

Recall the formula for the tangent of double the angle: \(\tan (2 \theta) = \frac{2 \tan \theta}{1 - \tan^{2} \theta}\). This equation will be proven true by simplifying the left and right side of the equation separately.

Step 2: Simplify the Left Side of the Equation

The left side of the equation is already simplified as \(\tan 2\theta\). There's no more simplification needed.

Step 3: Simplify the Right Side of the Equation

The right side of the equation is already the result of applying the tangent double angle formula: \(\frac{2 \tan \theta}{1 - \tan^{2} \theta}\). No simplification needed.

Step 4: Compare Both Sides of the Equation

After simplification, both sides of the equation are the same: \(\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^{2} \theta}\). This verifies the identity.

Key Concepts

Tangent Double Angle Formula

In trigonometry, the tangent double angle formula is one of the essential identities used to deal with angles that are twice as large as a given angle. This formula is expressed as:\[ \tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta} \] This identity is derived from the tangent sum identity, which is given by:\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]By substituting \(A = B = \theta\), we can express the sum identity for double angles:\[ \tan(\theta + \theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta} \].This identity is fundamental, particularly in calculus and physics, where it simplifies problems involving rotational or oscillatory systems. By understanding and applying this formula, you can more easily address problems concerning double angles in trigonometry.

Algebra

Algebra is a branch of mathematics that operates primarily on symbols and variables rather than numbers alone. In the context of trigonometric identities, algebra is crucial when it comes to simplifying expressions.

• It allows us to manipulate and rearrange equations.
• Algebraic operations include addition, subtraction, multiplication, division, and factoring.
• With algebra, we can transform complex trigonometric expressions into simpler forms.

So, in proving the tangent double angle identity, algebra helps in expressing the tangent of a double angle in a more simplified form. By comparing both sides of an identity (already simplified as in our example), you can verify its validity. Mastering these algebraic manipulations is vital for working through trigonometric identities successfully.

Angle Sum Identity

The angle sum identity is a collection of important trigonometric formulas that express the trigonometric functions of the sum of two angles in terms of the functions of the individual angles. One key formula is for tangent:\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]This identity forms the basis for the tangent double angle formula.
The beauty of the angle sum identity lies in its ability to connect different trigonometric functions. By adding angles, we can find precise values for otherwise challenging expressions. Whether simplifying expressions or solving equations, recognizing these identities helps manage a variety of trigonometric problems easily and effectively.These identities are highly useful in both theoretical math and practical applications, such as physics or engineering, where angle manipulation is often needed. Understanding these concepts is essential for anyone studying trigonometry at a deeper level.