Problem 31 - Algebra and Trigonometry

Question

27–32 Simplify the expression by using a double-angle formula or a half-angle formula. (a) \(\frac{\sin 8^{\circ}}{1+\cos 8^{\circ}} \quad\) (b) \(\frac{1-\cos 4 \theta}{\sin 4 \theta}\)

Step-by-Step Solution

Answer

(a) Simplifies to \(\tan 4^{\circ}\); (b) simplifies to \(\tan 2\theta\).

Step 1: Choosing the Right Formula for (a)

We need to simplify \( \frac{\sin 8^{\circ}}{1+\cos 8^{\circ}} \). We will use the identity for the tangent of a half-angle: \( \tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{\sin\theta} \). However, we can rearrange this as: \( \tan\left(\frac{\theta}{2}\right) = \frac{\sin\theta}{1 + \cos\theta} \).

Step 2: Simplifying Expression (a)

Recognize that the expression \( \frac{\sin 8^{\circ}}{1 + \cos 8^{\circ}} \) matches the form of \( \tan\left(\frac{8^{\circ}}{2}\right) \). Hence it simplifies directly to \( \tan 4^{\circ} \).

Step 3: Choosing the Right Formula for (b)

We need to simplify \( \frac{1 - \cos 4\theta}{\sin 4\theta} \). Once again, using the tangent of half angle identity, \( \tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{\sin\theta} \), this expression simplifies to \( \tan 2\theta \) where \( \theta = 4\theta \).

Step 4: Simplifying Expression (b)

Now \( \frac{1 - \cos 4\theta}{\sin 4\theta} \) simplifies to \( \tan 2\theta \) thus \( \tan 2\times 2\theta = \tan 2\theta \). Therefore, the expression is already in its simplest form as \( \tan 2\theta \).

Key Concepts

Double-Angle Formulas

Double-angle formulas are a type of trigonometric identity that relate functions of an angle to functions of its double, or twice its measure. They are especially handy when simplifying trigonometric expressions or solving trigonometric equations.
One of the most commonly used double-angle formulas is:

• \( \sin(2\theta) = 2 \sin\theta \cos\theta \)
• \( \cos(2\theta) = \cos^2\theta - \sin^2\theta \)
• \( \tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}\)

These formulas are useful because they allow us to express trigonometric functions of twice an angle in terms of the original angle. This is beneficial for simplifying expressions or solving equations involving trigonometric functions.

Half-Angle Formulas

Half-angle formulas are another group of trigonometric identities useful for breaking down expressions involving angles that are halved. These formulas help simplify expressions by relating the trigonometric functions of an angle to those of half the angle.
Here are some key half-angle formulas:

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

Each identity comes with a plus-minus sign, indicating depending on the angle's quadrant the outcome will switch between positive and negative. These formulas are instrumental in rewriting and simplifying trigonometric expressions by converting them into expressions of half angles.

Trigonometric Simplification

Trigonometric simplification involves rewriting complex trigonometric expressions into simpler or more manageable forms. This is particularly useful when working with equations that are challenging to solve directly.
In many cases, trigonometric identities, like the double-angle and half-angle formulas, are used to achieve simplification. The process typically involves:

• Identifying which identities can reduce complexity.
• Applying these identities to convert the original expression.
• Further simplifying using algebraic manipulations or additional identities.

For example, using the half-angle formula, \( \tan\left(\frac{\theta}{2}\right) = \frac{\sin\theta}{1 + \cos\theta} \), we can simplify the expression \( \frac{\sin 8^{\circ}}{1 + \cos 8^{\circ}} \) down to \( \tan 4^{\circ} \). Simplifying trigonometric expressions this way aids in solving equations and understanding trigonometric properties better.