Problem 26 - A Graphical Approach to Precalculus with Limits

Question

Use an identity to write each expression as a single trigonometric function or as a single number in exact form. Do not use a calculator. $$\frac{1}{8} \sin 29.5^{\circ} \cos 29.5^{\circ}$$

Step-by-Step Solution

Answer

\( \frac{1}{16} \sin 59^{\circ} \)

Step 1: Recall the Double Angle Identity for Sine

The double angle identity of sine states that \( an(2x) = 2 \sin(x) \cos(x) \). This identity states that twice the product of \( \sin \theta \) and \( \cos \theta \) is equal to the sine of the double angle.

Step 2: Convert the Given Expression Using the Identity

Our expression is \( \frac{1}{8} \sin 29.5^{\circ} \cos 29.5^{\circ} \). Using the identity, we know \( 2 \sin 29.5^{\circ} \cos 29.5^{\circ} = \sin(59^{\circ}) \). Hence, \( \sin 29.5^{\circ} \cos 29.5^{\circ} = \frac{1}{2} \sin(59^{\circ}) \).

Step 3: Simplify the given expression

Substitute the expression \( \sin 29.5^{\circ} \cos 29.5^{\circ} \) in terms of \( \sin(59^{\circ}) \) back to the original equation: \( \frac{1}{8} \sin 29.5^{\circ} \cos 29.5^{\circ} = \frac{1}{8} \times \frac{1}{2} \sin(59^{\circ}) \).

Step 4: Calculate Final Result

Simplify the expression, \( \frac{1}{8} \times \frac{1}{2} = \frac{1}{16} \). Therefore, the converted expression is \( \frac{1}{16} \sin 59^{\circ} \).

Key Concepts

Double Angle Identity

The Double Angle Identity is a powerful tool in trigonometry that helps us relate angles and trigonometric functions. One specific form of it, which relates to the sine function, is the equation: \( \sin(2x) = 2 \sin(x) \cos(x) \). This identity may seem complicated at first glance, but it is quite practical.

• It essentially shows how the sine of twice an angle can be expressed using the sine and cosine of the single angle.
• This identity is useful in simplifying expressions and solving equations where angles appear in doubled forms.
• It can help us understand the relationship between trigonometric functions and their angles more clearly.

In practice, if you have an expression involving \( \sin \theta \) and \( \cos \theta \), like in our exercise, applying this identity can often help reduce the complexity of the problem.

Sine Function

The sine function is one of the fundamental functions in trigonometry, characterized by its periodic and oscillating nature. It is crucial in the study of waves, circles, and oscillations. At its core:

• The sine function takes an angle and provides the length of the opposite side in a right triangle, standardized against the hypotenuse.
• Its values range between -1 and 1.
• Graphically, \( \sin(x) \) produces a smooth, continuous wave-like pattern that repeats every \( 360^\circ \) (or \( 2\pi \) radians).

When applying the sine function in trigonometric identities like the Double Angle Identity, it permits the transformation of complex expressions into simpler forms. This conversion is crucial when dealing with problems requiring exact values without the use of calculators.

Exact Values

Finding exact values in trigonometry means determining values for trigonometric functions that are not approximated but are precise. This task often requires understanding trigonometric identities and relationships, like the ones involved in our exercise.

• Exact values are necessary for calculations where precision is crucial, such as in mathematical proofs or engineering tasks.
• They can often be expressed in radical form, such as \( \sqrt{3}/2 \), instead of decimals (e.g., approximately 0.866).
• Using identities like the Double Angle Identity, complex expressions can be evaluated exactly, providing insights without computational rounding errors.

In solving our problem, using the Double Angle Identity allowed for the expression to be simplified to \( \frac{1}{16} \sin 59^\circ \), providing an exact form that demonstrates complete and precise representation of the original expression.