Problem 81 - Precalculus : Building Concepts and Connections

Question

Find the exact value of each expression without using a calculator. $$3 \sin \frac{\pi}{4}+2 \cos \frac{3 \pi}{4}$$

Step-by-Step Solution

Answer

The short answer is \( \sqrt{2}/2 \)

Step 1: Use the exact values of sine and cosine for the standard angles.

The standard value of sin(pi/4) = cos(pi/4) = \( \sqrt{2}/2 \), since sine and cosine angles are equal for 45 degrees. Similarly, for the standard value of cos(3pi/4) = -cos(pi/4) = \( -\sqrt{2}/2 \), the negative sign because cosine is negative for an angle in the second quadrant where 135 degrees is.

Step 2: Substitute the trigonometric values into the original expression.

The expression becomes: \(3 * \sqrt{2}/2 + 2 * ( -\sqrt{2}/2 )\)

Step 3: Simplify the result.

This gives us: \(3\sqrt{2}/2 - \sqrt{2} = \sqrt{2}/2\)

Key Concepts

Exact Values of Trigonometric Functions

Finding exact values for trigonometric functions often involves standard angles such as \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \) and \(\frac{\pi}{2}\). These angles correspond to various degrees like 0, 30, 45, 60, and 90. At these specific points, trigonometric ratios have values that can be expressed in exact mathematical terms.
For instance, at \(\frac{\pi}{4}\), both sine and cosine share the value \( \frac{\sqrt{2}}{2} \). Knowing these allows you to solve complex expressions quickly without needing a calculator.

• These values are derived from key geometric principles like the unit circle and Pythagorean identities.
• When manipulating expressions, understanding these values helps simplify and find solutions accurately.

Sine and Cosine Functions

Sine and cosine are fundamental to trigonometry, representing ratios of sides in right triangles. The sine of an angle is the ratio of the opposite side to the hypotenuse, while cosine is the adjacent side over the hypotenuse.
These functions repeat their values in a specific pattern, known as periodicity. For instance, both sine and cosine have a period of \(2\pi\), meaning their values recur every full circle.

• These functions are essential for determining distances, angles, and even are used in modeling wave patterns.
• Sine and cosine are complementary, with the relationship \( \sin(\theta) = \cos(\frac{\pi}{2} - \theta) \).

Understanding this can massively help in solving problems involving trigonometric expressions, enabling simpler substitutions and calculations.

Quadrants in Trigonometry

The coordinate plane is divided into four quadrants, affecting the sign of trigonometric functions. This is because each function has different representations in each quadrant.
To understand this better:

• The first quadrant (0 to \(\frac{\pi}{2}\)) sees all trigonometric functions positive.
• In the second quadrant (\(\frac{\pi}{2}\) to \(\pi\)), sine is positive, while cosine is negative, as used in the solution case \( \cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} \).
• The third quadrant (\(\pi\) to \(\frac{3\pi}{2}\)) has both sine and cosine negative.
• Finally, the fourth quadrant (\(\frac{3\pi}{2}\) to \(2\pi\)) sees sine negative and cosine positive.

These rules help in understanding the behavior of trigonometric functions across different angles, which is crucial for accurate calculations and solutions.