Problem 12 - Algebra 2

Question

Find the exact value of each expression. \(\cos 105^{\circ}\)

Step-by-Step Solution

Answer

\(\cos 105^{\circ} = \frac{\sqrt{2} - \sqrt{6}}{4}\).

Step 1: Express 105° as a Sum

We can express the angle 105° as a sum of two known angles whose trigonometric values we can easily find. Specifically, consider 105° as the sum of 60° and 45°, i.e., \(105° = 60° + 45°\).

Step 2: Apply the Cosine Addition Formula

The cosine addition formula is \(\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\). Apply this formula using \(a = 60°\) and \(b = 45°\).

Step 3: Substitute Known Values

Substitute the known trigonometric values into the addition formula:\(\cos(60°) = \frac{1}{2}\), \(\sin(60°) = \frac{\sqrt{3}}{2}\), \(\cos(45°) = \frac{\sqrt{2}}{2}\), \(\sin(45°) = \frac{\sqrt{2}}{2}\). The expression becomes \(\cos(105°) = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right)\).

Step 4: Perform Arithmetic Operations

Calculate each part of the expression: \(\frac{1}{2} \times \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4}\) and \(\frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2} = \frac{\sqrt{6}}{4}\). So we have \(\cos(105°) = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}\).

Step 5: Simplify the Expression

Combine the terms: \(\cos(105°) = \frac{\sqrt{2} - \sqrt{6}}{4}\). This is the exact value of the cosine of 105°.

Key Concepts

Angles as a Sum

When solving trigonometric problems, breaking down angles into sums of angles with known trigonometric values is often very helpful. For the angle 105°, you can express it as the sum of 60° and 45°. This approach utilizes angles that are commonly found on the unit circle or in trigonometric tables.

• This simplification allows the use of fundamental trigonometric formulas.
• Angles like 60° and 45° have easily memorizable cosine and sine values, which can be used efficiently.

By summing angles, you transform harder problems into simpler pieces. This method is not only effective for cosine but also for other trigonometric functions like sine and tangent. Understanding this concept lays the groundwork for applying various trigonometric formulas.

Trigonometric Values

Trigonometric values of certain angles, such as 45° and 60°, are fundamental in solving problems involving sums of angles. These values can often be determined using basic trigonometric principles.

• For 60°, the cosine value is \(\cos(60°) = \frac{1}{2}\), and the sine value is \(\sin(60°) = \frac{\sqrt{3}}{2}\).


• For 45°, both cosine and sine share the value \(\cos(45°) = \sin(45°) = \frac{\sqrt{2}}{2}\).

These precise trigonometric values lead to exact solutions and are frequently utilized in various mathematical fields including physics and engineering. Becoming familiar with these values can significantly speed up problem-solving. In more complex scenarios, they form the basis for computations involving compound angles.

Arithmetic Operations

Arithmetic operations play a crucial role once you substitute the known trigonometric values into an expression. After applying the cosine addition formula, you'll perform multiplication and subtraction.

• First, multiply the trigonometric values: \(\frac{1}{2} \times \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4}\) and \(\frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2} = \frac{\sqrt{6}}{4}\).
• Next, subtract the results: \(\frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}\).

The operations reduce to a simple form: \(\cos(105°) = \frac{\sqrt{2} - \sqrt{6}}{4}\). Understanding and executing these arithmetic steps accurately is essential to reach the exact trigonometric value. Mastering this process enhances precision in mathematical calculations and fosters a deeper understanding of how theoretical formulas apply in real-world contexts.