Problem 29 - Algebra and Trigonometry

Question

\(29-34\) Simplify the expression by using a Double-Angle Formula or a Half- Angle Formula. $$ \begin{array}{ll}{\text { (a) } 2 \sin 18^{\circ} \cos 18^{\circ}} & {\text { (b) } 2 \sin 3 \theta \cos 3 \theta}\end{array} $$

Step-by-Step Solution

Answer

(a) \(\sin 36^{\circ}\); (b) \(\sin 6\theta\).

Step 1: Identify the Formula to Use

The expression involves terms like \(2 \sin A \cos A\). This is a signal to use the double-angle formula for sine: \(2 \sin A \cos A = \sin 2A\).

Step 2: Apply the Formula to Part (a)

For expression \(2 \sin 18^{\circ} \cos 18^{\circ}\), recognize it as \(\sin 2A\) where \(A = 18^{\circ}\). Hence, \(\sin 2 \times 18^{\circ} = \sin 36^{\circ}\).

Step 3: Apply the Formula to Part (b)

For expression \(2 \sin 3\theta \cos 3\theta\), recognize it as \(\sin 2A\) where \(A = 3\theta\). Hence, \(\sin 2 \times 3\theta = \sin 6\theta\).

Step 4: Conclude the Simplified Expressions

The simplified expression for part (a) is \(\sin 36^{\circ}\) and for part (b) is \(\sin 6\theta\).

Key Concepts

Double-Angle Formula

The double-angle formula is an essential tool in trigonometry. It helps to simplify expressions involving trigonometric functions. In particular, the formula for sine is given by the expression \[ 2 \sin A \cos A = \sin 2A \]. This formula allows us to transform a product of sine and cosine into a single sine function with double the angle.

To apply the double-angle formula, you identify parts of an expression that match the left side of the formula. For instance, in the expression \( 2 \sin 18^{\circ} \cos 18^{\circ} \), it aligns perfectly with \( 2 \sin A \cos A \). Therefore, using the formula, we can simplify it to \( \sin 36^{\circ} \), because \( A \) is \( 18^{\circ} \), making \( 2A \) equal to \( 36^{\circ} \).

This method is particularly useful for reducing complexity in calculus or geometry problems involving trigonometric terms.

Half-Angle Formula

Half-angle formulas in trigonometry allow us to express trigonometric functions of half an angle in terms of the function value of the original angle. These formulas are beneficial in deriving values of trigonometric functions without a calculator.

For sine, the half-angle formula is written as:\[ \sin \frac{A}{2} = \pm \sqrt{\frac{1 - \cos A}{2}} \]. For cosine, it is:\[ \cos \frac{A}{2} = \pm \sqrt{\frac{1 + \cos A}{2}} \].

The sign in the formula depends on the quadrant where the half-angle \( \frac{A}{2} \) is located. It's important to check the original angle's location on the trigonometric circle to select the correct sign.

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive; cosine is negative.
• Quadrant III: Both are negative.
• Quadrant IV: Sine is negative; cosine is positive.

Half-angle formulas are particularly advantageous when angles form a special angle situation, like in the geometry of triangles or angular transformations.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is about rewriting a complex trigonometric function so that it's easier to understand or compute. This often involves using trigonometric identities, like the double-angle and half-angle formulas, to transform and reduce terms.

There are key steps to consider when simplifying:

• Identify and apply relevant formulas: Recognize patterns in the expression that fit known trigonometric identities.
• Factor and combine like terms: Simplify any algebraic terms or combine similar trigonometric terms.
• Check the domain and range: Ensure that the simplified expression is valid over the intended domain.

For instance, when faced with \( 2 \sin 3\theta \cos 3\theta \), we can simplify it using the double-angle formula, transforming it into \( \sin 6\theta \). This reduces a potentially lengthy calculation into a neat, single expression, which is much more manageable in problem-solving contexts.

Mastering simplification is a fundamental skill for handling more advanced trigonometric identities and solving trig-related problems efficiently.