Problem 86 - Precalculus

Question

For the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth decimal point. \(\cos \left(165^{\circ}\right)\)

Step-by-Step Solution

Answer

The exact value of \(\cos(165^{\circ})\) is \(-\frac{\sqrt{2} + \sqrt{6}}{4}\), approximately -0.9659.

Step 1: Use the Cosine Addition Formula

To find the exact value of \(\cos(165^{\circ})\), we can express it as \(165^{\circ} = 120^{\circ} + 45^{\circ}\). We will use the cosine addition formula: \[\cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b)\] Let \(a = 120^{\circ}\) and \(b = 45^{\circ}\).

Step 2: Find the Cosine and Sine Values

We know the cosine and sine values for \(120^{\circ}\) and \(45^{\circ}\):\(\cos(120^{\circ}) = -\frac{1}{2}, \quad \sin(120^{\circ}) = \frac{\sqrt{3}}{2} \)\(\cos(45^{\circ}) = \frac{\sqrt{2}}{2}, \quad \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \)

Step 3: Substitute into the Formula

Substituting these values into the cosine addition formula, we have:\[\cos(165^{\circ}) = \left(-\frac{1}{2}\right) \left(\frac{\sqrt{2}}{2}\right) - \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2}}{2}\right)\]

Step 4: Simplify the Expression

Simplify the expression:\[\cos(165^{\circ}) = -\frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}\]Combine the terms:\[\cos(165^{\circ}) = -\frac{\sqrt{2} + \sqrt{6}}{4}\]

Step 5: Verify the Answer Using a Calculator

Using a calculator, find the decimal approximation of \(\cos(165^{\circ})\). The result should be approximately -0.9659.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations relating various trigonometric functions. They are essential tools in geometry, physics, engineering, and various fields. A deep understanding of them allows us to simplify complex trigonometric expressions and solve equations. One of the fundamental identities is the **Pythagorean identity**: - For any angle \( \theta \), \( \sin^2(\theta) + \cos^2(\theta) = 1 \). This identity relates sine and cosine functions and forms the basis for deriving numerous other identities.
- **Reciprocal identities** help define trigonometric functions in terms of one another. For instance, \( \sec(\theta) = \frac{1}{\cos(\theta)} \) and \( \csc(\theta) = \frac{1}{\sin(\theta)} \).Understanding and applying these identities can greatly ease the process of evaluating trigonometric functions like cosine or sine of angles that are not commonly found in standard trigonometric tables.

Angle Addition

The angle addition formulas help calculate the trigonometric function of an angle that is the sum of two more common angles. In the case of cosine, the formula is: \[ \cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b) \]This formula allows us to express the trigonometric functions of a sum of angles in terms of the trigonometric functions of the individual angles. For example, to find \( \cos(165^{\circ}) \), we used the sum \( 165^{\circ} = 120^{\circ} + 45^{\circ} \).
- Identify parts \( a \) and \( b \).
- Find the values of \( \cos(a) \), \( \cos(b) \), \( \sin(a) \), and \( \sin(b) \).
- Substitute these values into the cosine addition formula to solve.This approach transforms a challenging problem into a manageable one by breaking down the calculation into smaller, more familiar parts.

Exact Values in Trigonometry

Exact values in trigonometry refer to the precise evaluation of trigonometric functions, often using radians or degrees such as \( \pi, \frac{\pi}{2}, \frac{\pi}{4} \), and their multiples. These values help replace decimals when solving analytical problems. Key angles like \( 30^{\circ}, 45^{\circ}, \text{and } 60^{\circ} \) often show up, having these simple value pairs:- \( \cos(45^{\circ}) = \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \)- \( \cos(120^{\circ}) = -\frac{1}{2} \)- \( \sin(120^{\circ}) = \frac{\sqrt{3}}{2} \)Recognizing these exact values allows for precise calculations without relying on approximations.
If we didn't use exact values, determining \( \cos(165^{\circ}) \) would be less accurate and more cumbersome. These values are crucial when deriving solutions using fundamental trigonometric identities and angle addition formulas.