Problem 30 - Algebra and Trigonometry

Question

\(29-34\) Simplify the expression by using a Double-Angle Formula or a Half- Angle Formula. $$ \begin{array}{ll}{\text { (a) } \frac{2 \tan 7^{\circ}}{1-\tan ^{2} 7^{\circ}}} & {\text { (b) } \frac{2 \tan 7 \theta}{1-\tan ^{2} 7 \theta}}\end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Formula Needed

To simplify the expression, we need to recall the double-angle identity for tangent. This identity states that \( \tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta} \). This means that the given expression is already in the form of this identity.

Step 2: Simplify Expression (a)

For part (a), recognize that the given expression \( \frac{2 \tan 7^{\circ}}{1 - \tan^2 7^{\circ}} \) matches the double-angle formula \( \tan(2\theta) \) where \( \theta = 7^{\circ} \). Therefore, \( \tan(2 \times 7^{\circ}) = \tan(14^{\circ}) \).

Step 3: Simplify Expression (b)

For part (b), use the same recognition as in step 2. The expression \( \frac{2 \tan 7\theta}{1 - \tan^2 7\theta} \) matches \( \tan(2\theta) \) with \( \theta = 7\theta \). Therefore, it simplifies directly to \( \tan(14\theta) \).

Key Concepts

Half-Angle Formula

The Half-Angle Formula is a set of trigonometric identities that allows us to find the trigonometric functions of half-angles. These formulas are particularly useful when solving problems involving angles that aren't directly measurable or easily calculable. The primary half-angle formulas are as follows for sine, cosine, and tangent:

• \( \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}} \)
• \( \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}} \)
• \( \tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{\sin(\theta)} = \frac{\sin(\theta)}{1 + \cos(\theta)} \)

These identities are derived from the double-angle formulas and solve half-angle problems efficiently. The sign (positive or negative) in the square roots depends on the quadrant in which the resulting angle lies. Each half-angle formula is related to its corresponding double-angle formula. This close connection makes them very powerful tools in solving trigonometric simplifications.

tangent

Tangent is one of the fundamental trigonometric functions and is particularly unique due to its interesting properties, like periodicity and its behavior at specific angles. The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the adjacent side. This can be represented as:

• \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)
• It also equals the sine of an angle divided by the cosine of the same angle: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

Tangent functions have a period of \( \pi \) radians (or 180 degrees), meaning that \( \tan(\theta) = \tan(\theta + n\pi) \) for any integer \( n \). Understanding tangent is essential when working with double and half-angle identities, as shown in the exercise where a double-angle formula for tangent is used.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that hold true for every value of the variable(s). These identities are crucial in simplifying and solving trigonometric expressions. They include basic Pythagorean identities such as:

• \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• \( 1 + \tan^2(\theta) = \sec^2(\theta) \)

Additionally, there are useful identities for angles and their multiples, like the angle addition formulas:

• \( \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) \)
• \( \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) \)
• \( \tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a)\tan(b)} \)

The double-angle and half-angle formulas are extensions of these identities. Mastery of trigonometric identities is fundamental for students to proficiently manipulate and resolve trigonometry problems, including simplifying complex expressions like those addressed in the original exercise.