Problem 83 - Precalculus : Building Concepts and Connections

Question

Find the exact value of each expression without using a calculator. $$\sin \frac{\pi}{4} \cos \frac{\pi}{4}$$

Step-by-Step Solution

Answer

The exact value of the given expression is \(\frac{1}{2}\).

Step 1: Recognize standard trigonometric values

For an angle of \(\frac{\pi}{4}\) radians, or 45 degrees, the sine and the cosine are both equal to \(\frac{\sqrt{2}}{2}\). This can be proven geometrically or by using the unit circle, and it's also a standard value that you often remember.

Step 2: Substitute sine and cosine values

So we can replace \(\sin \frac{\pi}{4}\) and \(\cos \frac{\pi}{4}\) in our original expression with \(\frac{\sqrt{2}}{2}\). This makes the expression \(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}\).

Step 3: Simplify the expression

Now, simplifying the expression, we get \((\frac{\sqrt{2}}{2})^2 = \frac{2}{4} = \frac{1}{2}\). The square of \(\sqrt{2}/2\) is \(\frac{2}{4}\), which reduces to \(\frac{1}{2}\).

Key Concepts

Sine and Cosine Functions

The sine and cosine functions are fundamental components in trigonometry. They relate the angles of a Right-Angled Triangle to the ratios of its sides. Here's why they are indispensable:

• **Sine (\(\sin \theta\))**: Represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.
• **Cosine (\(\cos \theta\))**: Represents the ratio of the length of the adjacent side to the hypotenuse.

These functions are not limited to only angles in triangles; they also apply to radian measures which come in handy with the unit circle—an essential tool for understanding these functions.
For the angle \(\frac{\pi}{4}\) radians (equivalent to 45 degrees), using the unit circle, both sine and cosine values are equal and commonly remembered as \(\frac{\sqrt{2}}{2}\). This is a standard value you will encounter often in trigonometry.
This equal value arises because, at 45 degrees, the opposite and adjacent sides of the triangle are of equal length when derived from a circle with a radius of 1 unit.

Radians to Degree Conversion

When dealing with trigonometric functions, it is common to encounter angles in both degrees and radians. Understanding the conversion between these two is key. The conversion formula is:\[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]Let's convert radians to degrees with an example:

• For \(\frac{\pi}{4}\) radians, multiplying by \(\frac{180}{\pi}\) gives you 45 degrees.

This conversion is vital because certain trigonometric values are easier to remember in degrees, while many mathematical fields prefer radians, especially in calculus and physics.
Remember that \(\pi\) radians equals 180 degrees, which is the backbone of converting between these two units.

Simplifying Expressions

Simplifying expressions in mathematics is about reducing them to their simplest form. This often involves combining like terms or using standard algebraic operations. Here's how we simplify trigonometric expressions:

• Start by substituting known standard values, such as \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\) and \(\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\).
• Next, multiply the values as given in the expression: \(\frac{\sqrt{2}}{2} \times \frac{\sqrt{2}}{2}\).
• This results in \(\left(\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} = \frac{1}{2}\).

Breaking it down, when you square \(\sqrt{2}\), you get 2, and when you square 2, you get 4. Hence, the expression results in the simplified fraction \(\frac{1}{2}\). This simplification process is crucial because it provides a quick, efficient way to handle complex expressions and aids in the verification of results.