Problem 54 - Algebra 2

Question

Use half-angle identities to write each expression, using trigonometric functions of \(\theta\) instead of \(\frac{\theta}{4} .\) $$ \sin \frac{\theta}{4} $$

Step-by-Step Solution

Answer

\(\sin\left(\frac{\theta}{4}\right) = \pm \sqrt{\frac{1-\cos\left(\frac{\theta}{2}\right)}{2}}\)

Step 1: Express the angle as double

Write \(\frac{\theta}{4}\) as \(\frac{\theta'}{2}\), where \(\theta'=\frac{\theta}{2}\). Now our task is to express \(\sin\left(\frac{\theta'}{2}\right)\) in terms of \(\theta\).

Step 2: Apply the half-angle identity for sine

Using the half-angle identity for sine, we have \(\sin\left(\frac{\theta'}{2}\right)= \pm \sqrt{\frac{1-\cos\theta'}{2}}\) . This gives us an expression for \(\sin\left(\frac{\theta}{4}\right)\) in terms of \(\frac{\theta}{2}\).

Step 3: Substitute back the angle

Replace \(\theta'\) back with \(\frac{\theta}{2}\) in the above expression to get our final result: \(\sin\left(\frac{\theta}{4}\right) = \pm \sqrt{\frac{1-\cos\left(\frac{\theta}{2}\right)}{2}}\)

Key Concepts

Trigonometric Functions

Trigonometric functions are like tools in mathematics that help us understand and work with angles and triangles. These functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), among others. Each has its own special way of dealing with angles, but they all relate to circles, since trigonometry often involves circles and circular motion.

Think of working with a circle: the circumference helps define our angles, and each point on the circle corresponds to an angle. Trigonometric functions help us find important information about this angle, like the heights or distances at different points in our circle.

• \( \sin \theta \) gives the vertical height for an angle \( \theta \).
• \( \cos \theta \) gives the horizontal distance for \( \theta \).
• \( \tan \theta \) is a ratio that combines both these values.

This is why we often use trigonometric functions in physics, engineering, and even computer graphics. They allow us to translate and transform angles into visual and functional information that can be further used in calculations.

Sine Function

The sine function is one of the most common functions in trigonometry, and it's all about measuring the vertical side of a right triangle when given an angle. It's especially useful when dealing with periodic phenomena, like waves or circular motion.

When we talk about the sine function, \( \sin \theta \), we refer to the ratio of the length of the side opposite the angle to the hypotenuse side of a right triangle. The value of sine can vary between \(-1\) and \(1\), depending on the angle.

To make calculations easier, there are certain identities that relate angles in different ways. One such identity is the half-angle identity:\[ \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos\theta}{2}} \]This formula can be used to find the sine of half of any angle, which is crucial in problems where angles are divided, like in this exercise. This way, you calculate \( \sin \frac{\theta}{4} \) in terms of \( \cos \theta \), offering a simplified expression, and avoiding direct measurements.

Angle Transformation

In trigonometry, angle transformation is about changing the way we express and use angles. This is useful for simplifying problems and making connections between different aspects of trigonometric functions.

For example, in the exercise, we transform \( \frac{\theta}{4} \) by expressing it as \( \frac{\theta'}{2} \) and later substituting it back to \( \frac{\theta}{2} \) in terms of \( \theta \).
This rearrangement allows us to use known identities like the half-angle identity, making complex angles more manageable.

Whether we're breaking down an angle into parts, combining different angles, or transforming them into a whole different expression, angle transformations help streamline the calculation process. It often reveals hidden relationships between angles that are not immediately apparent and can significantly simplify solving more complex equations and problems.