Problem 37 - Precalculus with Calculus Previews

Question

Use a half-angle formula to find the exact value of the given trigonometric function. Do not use a calculator. $$ \sin (3 \pi / 8) $$

Step-by-Step Solution

Answer

\( \sin(3\pi/8) = \frac{\sqrt{2 + \sqrt{2}}}{2} \).

Step 1: Identify the Half-Angle Formula

We'll use the half-angle formula for sine: \( \sin \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}} \). We need \( \cos(\theta) \) where \( \theta = \frac{3\pi}{4} \) because \( \displaystyle \frac{3\pi}{8} = \frac{1}{2} \cdot \frac{3\pi}{4} \).

Step 2: Find the Cosine of Related Angle

\( \cos\left(\frac{3\pi}{4}\right) \) is needed. From knowledge of the unit circle, \( \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \).

Step 3: Apply the Half-Angle Formula

Substitute \( \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \) into the half-angle formula:\[ \sin\left(\frac{3\pi}{8}\right) = \pm \sqrt{\frac{1 - \left(-\frac{\sqrt{2}}{2}\right)}{2}} = \pm \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}. \]

Step 4: Simplify the Expression

Simplify inside the square root:\[ \frac{1 + \frac{\sqrt{2}}{2}}{2} = \frac{2 + \sqrt{2}}{4}. \]Further simplify:\[ \sin\left(\frac{3\pi}{8}\right) = \pm \sqrt{\frac{2 + \sqrt{2}}{4}} = \pm \frac{\sqrt{2 + \sqrt{2}}}{2}. \]

Step 5: Determine the Correct Sign

Since \( \frac{3\pi}{8} \) is in the first quadrant where sine is positive, the positive value is appropriate. Thus:\[ \sin\left(\frac{3\pi}{8}\right) = \frac{\sqrt{2 + \sqrt{2}}}{2}. \]

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are a fundamental part of mathematics, especially in geometry. They help us relate angles to the lengths of sides in a right-angled triangle. There are six main trigonometric functions: sine (\( \sin \)), cosine (\( \cos \)), tangent (\( \tan \)), cosecant (\( \csc \)), secant (\( \sec \)), and cotangent (\( \cot \)). These functions are periodic, meaning they repeat their values in regular intervals.

The sine function, which measures the ratio of the opposite side to the hypotenuse in a right-angle triangle, is especially important. It varies between -1 and 1 and has a period of \( 2\pi \). The cosine and sine functions are closely related, often differing only by a phase shift of \( \frac{\pi}{2} \) radians. Understanding these relationships allows for the use of various identities and formulas, such as the half-angle formula, which can simplify complex trigonometric calculations.

Navigating the Unit Circle

The unit circle is a powerful tool in trigonometry. It's a circle with a radius of 1, centered at the origin of a coordinate plane. This circle provides a geometric representation of trigonometric functions, linking angle measures with coordinates. Each point on the unit circle has coordinates (\( \cos \theta, \sin \theta \)), representing angle \( \theta \) in standard position.

By knowing the unit circle well, you can directly find values for trigonometric functions without needing a calculator. For instance, the cosine of \( \frac{3\pi}{4} \) is found by looking at the unit circle. In the second quadrant, cosine is negative, leading to \( \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \). This knowledge is crucial for understanding how to simplify expressions using trigonometric formulas like the half-angle formula.

Calculating the Exact Value of Sine

Finding the exact value of a sine function without a calculator often involves using formulas and mathematical identities. One of these tools is the half-angle formula, which is useful when working with angles that aren't typically part of the standard angles on the unit circle. The half-angle formula for sine is represented as:\[ \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}} \]By applying this formula, you can break down the problem for the angle \( \frac{3\pi}{8} \), which is half of \( \frac{3\pi}{4} \).

Let's solve it! First, identify the cosine of the related angle. Knowing \( \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \), substitute this value into the half-angle formula:\[ \sin\left(\frac{3\pi}{8}\right) = \pm \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} \]Further simplification gets you to:\[ \sin\left(\frac{3\pi}{8}\right) = \pm \frac{\sqrt{2 + \sqrt{2}}}{2} \]Since \( \frac{3\pi}{8} \) is in the first quadrant, where the sine is positive, the exact value is:\[ \sin\left(\frac{3\pi}{8}\right) = \frac{\sqrt{2 + \sqrt{2}}}{2} \]This step-by-step use of identities shows how understanding the structure of trigonometric functions and their relationships helps solve otherwise complex problems with ease.