Question

Find the exact values in Hint: Half-angle identities may be helpful. $$ \sin ^{3} \frac{\pi}{6} $$

Short answer

The exact value is \( \frac{1}{8} \).

Step-by-step solution

Identify the Angle

First, determine the angle for which we need to find the sine. From the expression \( \sin^3 \frac{\pi}{6} \), the angle is \( \frac{\pi}{6} \) radians.

Calculate \( \sin \frac{\pi}{6} \)

The sine of \( \frac{\pi}{6} \) radians (or 30 degrees) is a known value. It is \( \frac{1}{2} \).

Cube the Sine Value

Using the result from Step 2, cube the sine value. So, you compute \( \left( \sin \frac{\pi}{6} \right)^3 = \left( \frac{1}{2} \right)^3 = \frac{1}{8} \).

Key concepts

Half-Angle Identities

Half-angle identities are trigonometric identities that provide a way to express trigonometric functions of half-angles in terms of the functions of the original angles. These identities are especially beneficial when dealing with calculations involving specific angles in trigonometry. For example, when you need to work out the sine, cosine, or tangent of an angle that is not readily available using the standard angle values.

• They simplify complex expressions involving trigonometric functions.
• They are derived from the double-angle formulas.

For sine, the half-angle identity is: \[ ext{If } heta = 2eta, \text{ then } \ \sin \theta = 2 \sin \beta \cos \beta.\]For this exercise, the identity helps us to better understand the properties of sine at different angles.

Sine Function

The sine function is one of the primary trigonometric functions. It's crucial in mathematics, especially in the study of periodic phenomena like waves. The sine of an angle is defined in a right triangle as the ratio of the opposite side to the hypotenuse.

• The sine function is periodic with a period of \(2\pi\) radians (or 360 degrees).
• It varies between -1 and 1.
• It's positive in the first and second quadrants of the unit circle.

In this problem, we are asked to find the sine of \( \frac{\pi}{6} \), which is a standard angle in trigonometry. To recall, \(\sin \frac{\pi}{6} \) is equivalent to \(\frac{1}{2}\), a value you often encounter in trigonometric calculations and should be memorized.

Radians

Radians offer an alternative to degrees for measuring angles, commonly used in modern mathematics because they provide a natural way to describe angles in terms of the radius of a circle. The conversion between degrees and radians is essential for any trigonometry student.

• \(1 \text{ radian} \approx 57.2958 \text{ degrees}\).
• \(\pi \text{ radians} = 180 \text{ degrees}\).

When you see \(\frac{\pi}{6} \), it denotes an angle equivalent to 30 degrees. Understanding how to switch between radians and degrees will help when solving problems in trigonometry or calculus. Radians simplify a lot of expressions and are used extensively in higher mathematics like calculus, where angle measurements often relate to properties of circles and periodic motion.

Cube of Sine Value

The problem we are solving involves finding the cube of the sine value of a specific angle. This might seem simple at first, but it’s an essential exercise to understand the effect of exponentiation on trigonometric values.

• To cube a sine value means raising it to the power of three.
• This operation will affect the range of the sine function.
• When cubing, remember that even small changes in the base can lead to significant results.

For the angle \( \frac{\pi}{6} \), we know that \(\sin \frac{\pi}{6} = \frac{1}{2}\). Cubing this value gives: \[\left( \frac{1}{2} \right)^3 = \frac{1}{8}\]This final value illustrates how trigonometric expressions can be manipulated and highlights the importance of understanding the properties of functions like sine when raised to powers.