Problem 62 - Algebra 2

Question

Which expression is an exact value for \(\sin 15^{\circ} ?\) \(\begin{array}{ll}{\text { A. } \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2} \cdot \frac{1}{2}} & {\text { B. } \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2} \cdot \frac{1}{2}}\end{array}\) C. \(\frac{\sqrt{2}}{2} \cdot \frac{1}{2}+\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} \quad\) D. \(\frac{\sqrt{2}}{2} \cdot \frac{1}{2}-\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}\)

Step-by-Step Solution

Answer

The exact value for \( \sin 15^{\circ} \) is given by expression B.

Step 1: Utilize the Identity

Let's start by utilizing the sine difference identity to express \( \sin 15^{\circ} \). According to this identity, \( \sin(a - b) = \sin a \cos b - \cos a \sin b \). Therefore, we can express \( \sin 15^{\circ} \) as \( \sin (45^{\circ} - 30^{\circ}) = \sin 45^{\circ} \cos 30^{\circ} - \cos 45^{\circ} \sin 30^{\circ} \).

Step 2: Substitute Known Values

Next, substitute the known values: \( \sin 45^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2} \), \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \), and \( \sin 30^{\circ} = \frac{1}{2} \). So, \( \sin 15^{\circ} = \left(\frac{\sqrt{2}}{2}\right) \times \left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right) \times \left(\frac{1}{2}\right) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} \).

Step 3: Compare with options

Now, compare this value with the given answer choices. Only option B equates to the calculated value of \( \sin 15^{\circ} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} \)

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry, representing the ratio of the length of the side opposite an angle to the hypotenuse in a right triangle. It is commonly abbreviated as "sin". In general, for a given angle \( \theta \), the sine function is expressed as \( \sin(\theta) \). Although this function originates from right-angled triangles, it also extends to real numbers, indicating its relevance in periodic functions and circular motion.

• The sine function is periodic with a period of \( 360^{\circ} \) or \( 2\pi\) radians.
• Its values range from \(-1\) to \(1\). Hence, knowing these extremes helps in quickly understanding the behavior of \( \sin(\theta) \).

Understanding the sine function is crucial for solving various trigonometric problems, many of which involve using identities and equations to find exact values.

Angle Difference Identity

The angle difference identity is a powerful tool used in trigonometry to simplify expressions involving trigonometric functions of angles. For the sine function, this identity states that \( \sin(a - b) = \sin a \cos b - \cos a \sin b \).

• This identity allows the calculation of a sine function for an angle expressed as the difference of two other angles for which the trigonometric values are known.
• For example, to find \( \sin 15^{\circ} \), you can use the angle difference identity as \( \sin(45^{\circ} - 30^{\circ}) \), which simplifies to \( \sin 45^{\circ} \cos 30^{\circ} - \cos 45^{\circ} \sin 30^{\circ} \).

This identity is immensely useful in providing exact trigonometric values for angles not commonly found on the basic unit circle chart.

Exact Trigonometric Values

Exact trigonometric values are specific values of trigonometric functions that are derived from well-known angles. These include angles like \( 0^{\circ} \), \( 30^{\circ} \), \( 45^{\circ} \), \( 60^{\circ} \), and \( 90^{\circ} \). Each of these angles has a corresponding sine, cosine, and tangent value that can be expressed as fractions involving square roots, providing more precise calculation results than decimal approximations.

• For instance, \( \sin 45^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2} \), making them equal at this angle.
• Similarly, \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \) and \( \sin 30^{\circ} = \frac{1}{2} \) are key exact values.

By plugging in these exact trigonometric values in identities like the angle difference identity, you can calculate sine, cosine, or tangent values for more complex angles, like \( 15^{\circ} \) in this example, without resorting to a calculator for approximation. This deepens understanding and offers precision in mathematical trigonometry applications.