Problem 36 - 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. $$ \sin (\pi / 8) $$

Step-by-Step Solution

Answer

\( \sin(\pi/8) = \frac{\sqrt{2 - \sqrt{2}}}{2} \)

Step 1: Identify the Half-Angle Formula

To solve for \( \sin(\pi/8) \), we use the half-angle formula for sine: \[\sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}}\]In this case, we need to find \( \sin(\frac{\pi}{4}) \) as \( \theta = \pi/4 \).

Step 2: Determine the Value of \(\cos(\pi/4)\)

We know that \( \cos(\pi/4) = \frac{\sqrt{2}}{2} \) because \( \pi/4 \) is equivalent to 45 degrees, for which both sine and cosine are \( \frac{\sqrt{2}}{2} \).

Step 3: Apply the Half-Angle Formula

Plug \( \cos(\pi/4) = \frac{\sqrt{2}}{2} \) into the half-angle formula:\[\sin(\frac{\pi}{8}) = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}}\]We choose the positive square root because \( \frac{\pi}{8} \) is in the first quadrant where sine is positive.

Step 4: Simplify the Expression

Simplify inside the square root:\[\frac{1 - \frac{\sqrt{2}}{2}}{2} = \frac{2}{2} - \frac{\sqrt{2}}{2} = \frac{2 - \sqrt{2}}{2} \times \frac{1}{2} = \frac{2 - \sqrt{2}}{4}\]

Step 5: Simplify Further to Find the Exact Value

Now take the square root:\[\sin(\frac{\pi}{8}) = \sqrt{\frac{2 - \sqrt{2}}{4}} = \frac{\sqrt{2 - \sqrt{2}}}{2}\]This is the exact value of \( \sin(\frac{\pi}{8}) \).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in the study of triangles and circles. These functions relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions include sine ( \( \sin \) ), cosine ( \( \cos \) ), and tangent ( \( \tan \) ). Each function serves a specific purpose, and each is useful in different mathematical contexts. For example:

• Sine ( \( \sin \) ) measures the ratio of the opposite side to the hypotenuse in a right-angled triangle.
• Cosine ( \( \cos \) ) measures the ratio of the adjacent side to the hypotenuse.
• Tangent ( \( \tan \) ) measures the ratio of the opposite side to the adjacent side.

Trigonometric functions are used for calculations involving waves, circles, and oscillatory phenomena. Understanding these functions is essential in physics, engineering, and advanced mathematics. They are not only confined to angles expressed in degrees but can also apply to radian measures, as seen with our example with \( \pi/8 \).
Grasping trigonometric functions will provide a solid foundation for exploring more complex mathematical and real-world applications.

Exact Values

When working with trigonometric functions, it is often necessary to find exact values rather than decimal approximations. Exact values allow for precision in problems where approximations might lead to inaccurate results.
Given the trigonometric function \( \sin(\pi/8) \), we can find its exact value using established identities and formulas without relying on electronic calculators, which often return decimal numbers. This process includes knowing the exact values of commonly used angles, like 45 degrees (\( \pi/4 \)), where \( \cos(\pi/4) = \frac{\sqrt{2}}{2} \).
These exact values are generally derived from well-known trigonometric values that are simple fractions or \( \sqrt{} \) expressions. Using these values in conjunction with trigonometric identities allows us to evaluate expressions like \( \sin(\pi/8) \) without guesswork. Obtaining these exact values is vital when precision is necessary in mathematics and its applications.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. They are used to simplify complex trigonometric expressions and solve equations. The half-angle identities are a subgroup of these identities and are particularly useful for computing function values at specific angles.
The half-angle formula for sine is:\[ \sin \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}} \]This formula is derived from double-angle identities and helps us find the sine of half angles without direct measurement. It is especially handy with angles not readily calculated from known sums or differences, like \( \pi/8 \).
The half-angle identity for sine gives both a positive and negative root. Choosing the correct sign depends on the angle's quadrant: since the angle \( \pi/8 \) is in the first quadrant, where sine is positive, we use the positive root.
These identities reinforce our toolbox, helping solve trigonometric equations and find values precisely, building confidence in tackling a wide array of problems.