Problem 43 - Contemporary Precalculus

Question

Assume that $$\sin (\pi / 8)=\frac{\sqrt{2-\sqrt{2}}}{2}$$ and use identities to find the exact functional value. $$\cos (\pi / 8)$$

Step-by-Step Solution

Answer

Answer: $$\cos (\pi / 8) = \frac{\sqrt{2 + \sqrt{2}}}{2}$$

Step 1: Write down the given information

We know that $$\sin (\pi / 8) = \frac{\sqrt{2 - \sqrt{2}}}{2}$$.

Step 2: Use the Pythagorean identity

We use the Pythagorean identity to find the value of $$\cos (\pi / 8)$$. $$\sin^2(\pi / 8) + \cos^2(\pi / 8) = 1$$

Step 3: Substitute the given value of $$\sin (\pi / 8)$$

Substitute the given value of $$\sin (\pi / 8) = \frac{\sqrt{2 - \sqrt{2}}}{2}$$ into the equation. $$\left(\frac{\sqrt{2 - \sqrt{2}}}{2}\right)^2 + \cos^2(\pi / 8) = 1$$

Step 4: Simplify the equation

Now simplify the equation: $$\frac{2 - \sqrt{2}}{4} + \cos^2(\pi / 8) = 1$$

Step 5: Solve for $$\cos (\pi / 8)$$

To find the value of $$\cos (\pi / 8)$$, isolate it on one side of the equation: $$\cos^2(\pi / 8) = 1 - \frac{2 - \sqrt{2}}{4}$$

Step 6: Simplify the equation

Simplify the equation again: $$\cos^2(\pi / 8) = \frac{4 - (2 - \sqrt{2})}{4}$$ $$\cos^2(\pi / 8) = \frac{2 + \sqrt{2}}{4}$$

Step 7: Find the square root

To find the value of $$\cos (\pi / 8)$$, take the square root of both sides of the equation: $$\cos (\pi / 8) = \sqrt{\frac{2 + \sqrt{2}}{4}}$$ $$\cos (\pi / 8) = \frac{\sqrt{2 + \sqrt{2}}}{2}$$ Hence, the exact functional value of $$\cos (\pi / 8)$$ is $$\frac{\sqrt{2 + \sqrt{2}}}{2}$$.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a vital concept in trigonometry. It states that for any angle \( \theta \), the following equation holds: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] This formula helps relate the sine and cosine of an angle. To solve for one of these functions, you can rearrange the equation depending on what you already know.
For example, if the sine of an angle is known, then the cosine can be found by rearranging the identity to: \[ \cos^2(\theta) = 1 - \sin^2(\theta) \] You can then take the square root to solve for \( \cos(\theta) \).
This identity is derived from the Pythagorean Theorem by considering a right-angled triangle with hypotenuse of length 1.

Exact Trigonometric Values

Understanding exact trigonometric values can be crucial when it comes to solving trigonometry problems without a calculator. Certain angles such as \( \pi/8 \), \( \pi/4 \), \( \pi/3 \), and others, have values that can be exactly determined using algebraic expressions. These values are often tied to certain patterns or identities.
In the problem given, the exact value of \( \sin(\pi / 8) \) is provided as \( \frac{\sqrt{2 - \sqrt{2}}}{2} \). To find the exact value of \( \cos(\pi / 8) \) using the identities provides a deeper understanding of these relationships.
The importance of knowing these exact values lies in their ability to simplify complex expressions in calculus and other mathematical fields.

Angle Reduction Formula

Angle reduction formulas help in transforming trigonometric functions of specific angles into equivalent functions at simpler angles. These formulas are particularly useful when dealing with angles that don’t provide straightforward trigonometric values.
Consider solving for \( \cos(\pi/8) \) using revolution angles such as \( \pi/4 \) and identities. The sine and cosine values for these angles greatly assist in solving equations for angles like \( \pi/8 \). When \( \cos(\pi/8) \) was extracted using algebraic manipulation from the Pythagorean Identity, it exemplified the utility of these formulas.
If you have angles that are not commonly found in trigonometric tables, angle reduction may simplify the problem-solving process, making seemingly difficult angles manageable.