Problem 28 - Algebra and Trigonometry

Question

27–32 Simplify the expression by using a double-angle formula or a half-angle formula. (a) \(\frac{2 \tan 7^{\circ}}{1-\tan ^{2} 7^{\circ}} \quad\) (b) \(\frac{2 \tan 7 \theta}{1-\tan ^{2} 7 \theta}\)

Step-by-Step Solution

Answer

(a) \( \tan(14^\circ) \); (b) \( \tan(14\theta) \).

Step 1: Recognize the Identity

For both parts of the problem, observe that the expression \( \frac{2 \tan A}{1-\tan^2 A} \) is the double angle identity for tangent: \( \tan(2A) \). This identity helps simplify the given expression.

Step 2a: Apply the Identity to Part (a)

In the expression \( \frac{2 \tan 7^\circ}{1-\tan^2 7^\circ} \), assign \( 7^\circ \) as \( A \). Applying the identity, we have \( \tan(2 \times 7^\circ) = \tan(14^\circ) \). Thus, the expression simplifies to \( \tan(14^\circ) \).

Step 2b: Apply the Identity to Part (b)

Similarly, for the expression \( \frac{2 \tan 7\theta}{1-\tan^2 7\theta} \), assign \( 7\theta \) as \( A \). Applying the identity, we get \( \tan(2 \times 7\theta) = \tan(14\theta) \). Therefore, the expression simplifies to \( \tan(14\theta) \).

Key Concepts

Tangent Identity

The tangent identity is a crucial concept in trigonometry that simplifies expressions involving angles. One of the most powerful identities in this regard is the double-angle identity for tangent.
Given by the formula:

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

This identity allows us to transform a more complex tangent expression into a simpler one involving double the angle. It's highly useful in simplifying trigonometric expressions and solving equations.
In the given exercise, recognizing the structure of \( \frac{2 \tan A}{1-\tan^2 A} \) as the double angle identity for tangent is the key to solving the problem. This understanding helps us convert the expressions to \( \tan(14^\circ) \) and \( \tan(14\theta) \) by essentially doubling the angle from the original \( A \).
Using these identities, complex angle relationships become easier to manage and manipulate, simplifying calculations and providing clearer insights into trigonometric problems.

Trigonometric Simplification

Trigonometric simplification involves using identities and theorems to reduce complex expressions into simpler forms. This process makes working with trigonometric equations much easier. It also helps in recognizing patterns and further applying them to solve problems.
When tackling the given exercise, the simplification begins by identifying the expression that matches a known identity — in this case, the double-angle identity for tangent. By applying the correct identity:

• \( \frac{2 \tan 7^\circ}{1 - \tan^2 7^\circ} \) becomes \( \tan(14^\circ) \)
• \( \frac{2 \tan 7\theta}{1 - \tan^2 7\theta} \) becomes \( \tan(14\theta) \)

This approach streamlines otherwise elaborate expressions.Trigonometric simplification like this allows us to draw direct and quick conclusions without extensive calculations. Recognizing these identities early on can save effort and time while providing clear, concise solutions.

Angle Transformation

Angle transformation in trigonometry involves changing the representation of angles either by doubling, halving, or converting through identities. This skill is particularly essential when working with double-angle, half-angle, or sum and difference identities.
In the given exercise, we perform an angle transformation by using the double-angle formula. By taking the input angle (either \( 7^\circ \) or \( 7\theta \)) and doubling it:

• Transforms \( 7^\circ \rightarrow 14^\circ \)
• Transforms \( 7\theta \rightarrow 14\theta \)

These transformations leverage the tangent double-angle identity, simplifying the expression and thus making further calculations more straightforward.
Understanding angle transformations allows you to work seamlessly with different trigonometric functions and identities. It forms the backbone of solving more complex trigonometric equations, where a direct approach may be cumbersome or more challenging to work through.