Problem 33 - Algebra and Trigonometry

Question

\(29-34\) Simplify the expression by using a Double-Angle Formula or a Half- Angle Formula. $$ \begin{array}{ll}{\text { (a) } \frac{\sin 8^{\circ}}{1+\cos 8^{\circ}}} & {\text { (b) } \frac{1-\cos 4 \theta}{\sin 4 \theta}}\end{array} $$

Step-by-Step Solution

Answer

(a) \( \tan 4^{\circ} \) and (b) \( \tan 2\theta \)

Step 1: Identify Applicable Formula for Part (a)

For part (a), let's first focus on the expression \( \frac{\sin 8^{\circ}}{1+\cos 8^{\circ}} \). We'll use the identity: \( \sin \theta = 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} \) and \( 1+\cos \theta = 2 \cos^2 \frac{\theta}{2} \). These are derived from the double-angle formulas.

Step 2: Simplify Part (a) Using Double Angle Identities

Applying the identities from Step 1, we get:\[ \sin 8^{\circ} = 2 \sin 4^{\circ} \cos 4^{\circ} \] and\[ 1 + \cos 8^{\circ} = 2 \cos^2 4^{\circ} \]Substitute these into the expression: \[ \frac{2 \sin 4^{\circ} \cos 4^{\circ}}{2 \cos^2 4^{\circ}} = \frac{\sin 4^{\circ}}{\cos 4^{\circ}} = \tan 4^{\circ} \]

Step 3: Identify Applicable Formula for Part (b)

For part (b), the expression is \( \frac{1-\cos 4 \theta}{\sin 4 \theta} \). We can use the identity \( 1 - \cos \theta = 2 \sin^2 \frac{\theta}{2} \).

Step 4: Simplify Part (b) Using Half-Angle Identity

Applying the identity from Step 3, we have:\[ 1 - \cos 4\theta = 2 \sin^2 2\theta \] Substituting in the original expression, we get:\[ \frac{2 \sin^2 2\theta}{\sin 4\theta} = \frac{2 \sin^2 2\theta}{2 \sin 2\theta \cos 2\theta} = \frac{\sin 2\theta}{\cos 2\theta} = \tan 2\theta \]

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions to each other. These identities are crucial in simplifying complex trigonometric expressions by providing standard formulas that can be applied.

Some of the most commonly used identities include:

• Pythagorean Identities, like \( \sin^2\theta + \cos^2\theta = 1 \)
• Reciprocal Identities, such as \( \sin\theta = \frac{1}{\csc\theta} \)
• Double-Angle and Half-Angle Formulas, like \( \sin 2\theta = 2 \sin\theta \cos\theta \)

In the given problem, the focus is on Double and Half-Angle Formulas. These are key tools for breaking down expressions involving an angle into expressions involving half or double that angle, respectively. For instance, in step 1 of the solution, the double-angle formula for sine \( \sin \theta = 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} \) is used to simplify the numerator, \( \sin 8^{\circ} \), and \( 1 + \cos 8^{\circ} = 2 \cos^2 4^{\circ} \) simplifies the denominator.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions often involves using identities to rewrite expressions in a more manageable form. The goal is to convert complex fractions or products into more straightforward expressions.

By using known identities, you can break down a problem into smaller, more easily solvable parts. For part (a) of the original problem, the expression \[ \frac{\sin 8^{\circ}}{1+\cos 8^{\circ}} \] is simplified using the double-angle formulas.

Initially, these identities transform the expression into \[ \frac{2 \sin 4^{\circ} \cos 4^{\circ}}{2 \cos^2 4^{\circ}} \. \]This expression simplifies further to yield \( \tan 4^{\circ} \). A similar process happens in part (b), where the expression \[ \frac{1-\cos 4\theta}{\sin 4\theta} \] is tackled using half-angle identities to eventually simplify to \( \tan 2\theta \). Understanding how to rearrange and substitute using these identities is the core of simplifying any trigonometric problem.

Tan Function

The tangent function, denoted as tan, is one of the primary trigonometric functions. It is defined as the ratio of the sine and cosine of an angle: \( \tan\theta = \frac{\sin\theta}{\cos\theta} \). This function exhibits periodic behavior and has a period of \( \pi \) radians (or 180 degrees).

In the context of this problem, both parts (a) and (b) are simplified to involve the tangent function through their respective processes.

• Part (a) uses the identity relations to arrive at \( \tan 4^{\circ} \).
• Part (b) simplifies down to \( \tan 2\theta \).

Understanding the properties of the tan function, such as its asymptotic behavior and its use in right-angled triangle problems, is pivotal when working through trigonometry.

By recognizing when expressions can be reduced to a simple tangent function, complex problems are made significantly easier, making it a powerful tool in your trigonometry toolkit.