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Chapter 14: Problem 12

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

Short Answer

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

Step by step solution

01

Express 105掳 as a Sum

We can express the angle 105掳 as a sum of two known angles whose trigonometric values we can easily find. Specifically, consider 105掳 as the sum of 60掳 and 45掳, i.e., \(105掳 = 60掳 + 45掳\).
02

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 \(a = 60掳\) and \(b = 45掳\).
03

Substitute Known Values

Substitute the known trigonometric values into the addition formula:\(\cos(60掳) = \frac{1}{2}\), \(\sin(60掳) = \frac{\sqrt{3}}{2}\), \(\cos(45掳) = \frac{\sqrt{2}}{2}\), \(\sin(45掳) = \frac{\sqrt{2}}{2}\). The expression becomes \(\cos(105掳) = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right)\).
04

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 \(\cos(105掳) = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}\).
05

Simplify the Expression

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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掳, you can express it as the sum of 60掳 and 45掳. 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掳 and 45掳 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掳 and 60掳, are fundamental in solving problems involving sums of angles. These values can often be determined using basic trigonometric principles.
  • For 60掳, the cosine value is \(\cos(60掳) = \frac{1}{2}\), and the sine value is \(\sin(60掳) = \frac{\sqrt{3}}{2}\).

  • For 45掳, both cosine and sine share the value \(\cos(45掳) = \sin(45掳) = \frac{\sqrt{2}}{2}\).
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: \(\cos(105掳) = \frac{\sqrt{2} - \sqrt{6}}{4}\). 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.

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Most popular questions from this chapter

Verify that each of the following is an identity. $$ \sin ^{4} x-\cos ^{4} x=2 \sin ^{2} x-1 $$

Solve each equation for all values of \(\theta\) if \(\theta\) is measured in radians. \((\cos \theta)(\sin 2 \theta)-2 \sin \theta+2=0\)

Find the exact value of \(\sin 2 \theta, \cos 2 \theta, \sin \frac{\theta}{2},\) and \(\cos \frac{\theta}{2}\) for each of the following. \(\sin \theta=\frac{3}{5} ; 0^{\circ}<\theta<90^{\circ}\)

Find the exact values of \(\sin 2 \theta, \cos 2 \theta, \sin \frac{\theta}{2},\) and \(\cos \frac{\theta}{2}\) for each of the following. $$ \sin \theta=-\frac{1}{3} ; 270^{\circ}<\theta<360^{\circ} $$

Solve each equation for all values of \(\theta\). \(2 \cos ^{2} \theta+3 \sin \theta-3=0\)
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