Problem 27 - Algebra and Trigonometry

Question

27–32 Simplify the expression by using a double-angle formula or a half-angle formula. (a) 2 \(\sin 18^{\circ} \cos 18^{\circ} \quad\) (b) 2 \(\sin 3 \theta \cos 3 \theta\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Double-Angle Formula

Recognize that the expression \(2 \sin A \cos A\) can be simplified using the double-angle formula for sine: \(\sin(2A) = 2 \sin A \cos A\).

Step 2: Apply the Formula to Part (a)

For part (a), substitute \(A = 18^{\circ}\) into the formula. Hence, \(2 \sin 18^{\circ} \cos 18^{\circ} = \sin(2 \times 18^{\circ}) = \sin(36^{\circ})\).

Step 3: Apply the Formula to Part (b)

For part (b), use the same identity with \(A = 3\theta\). Therefore, \(2 \sin 3\theta \cos 3\theta = \sin(2 \times 3\theta) = \sin(6\theta)\).

Step 4: Verify the Results

Confirm that both simplifications use the double-angle identity correctly.For part (a), \(2 \sin 18^{\circ} \cos 18^{\circ}\) simplifies to \(\sin 36^{\circ}\).For part (b), \(2 \sin 3\theta \cos 3\theta\) simplifies to \(\sin 6\theta\).

Key Concepts

Double-Angle Formulas

The double-angle formulas are essential tools in trigonometry that simplify expressions involving trigonometric functions. Specifically, they express trigonometric functions of double angles in terms of single angles. A common formula, and the one used in the given exercise, is the double-angle formula for sine:\[\sin(2A) = 2 \sin A \cos A\]This identity provides a straightforward way to handle expressions like \(2 \sin A \cos A\) by converting it into a simpler single-angle sine term. You see this in action as it reduces the complexity of multi-term trigonometric expressions into just one term, as demonstrated by the solutions provided in the exercise. Double-angle formulas can be quite handy in solving trigonometric equations and simplifying trigonometric expressions encountered in various math problems.

Sine Function

The sine function is a fundamental trigonometric function, represented as \(\sin(\theta)\). It comes from the sides of a right triangle where \(\theta\) is an angle, defined as the ratio of the length of the opposite side to the hypotenuse. Some key aspects of the sine function include:

• Periodic Nature: Sine is periodic with a period of \(360^{\circ}\) or \(2\pi\) radians, meaning it repeats its values in each complete cycle.
• Range and Values: The range is from \(-1\) to \(1\), inclusive of these values.
• Symmetry: It is an odd function, implying that \( \sin(-\theta) = -\sin(\theta) \).

In the exercise, the double-angle formula applied to \(\sin A\) effectively blends these characteristics to simplify expressions with compound angles, showcasing the sine function's versatility in handling multiple angle identities and transformations.

Angle Simplification

Angle simplification in trigonometry involves the transformation of complex angle terms into simpler forms, often using trigonometric identities. By applying identities like double-angle formulas, more manageable expressions can be derived.When you simplify angles:

• Efficiency in Calculations: Reduction of complex multi-angle situations into easier computations is achieved.
• Clarity in Problem Solving: Simplified angles provide clearer insights during calculations, thus helping in solving trigonometric problems more effectively.
• Example in Action: In the exercise, \(2 \sin 18^{\circ} \cos 18^{\circ}\) is simplified to \(\sin 36^{\circ}\); this transformation illustrates how angle simplification reduces computational complexity.

Angle simplification not only aids in solving problems but also enhances understanding of trigonometric relationships by linking various functions and angles in an intuitive way.