Problem 14 - Algebra 2

Question

Find the exact value of each expression. \(\cos 165^{\circ}\)

Step-by-Step Solution

Answer

\(\cos 165^{\circ} = -\frac{\sqrt{2} + \sqrt{6}}{4}\)

Step 1: Use the identity for cosine of sum angles

To find the exact value of \(\cos 165^{\circ}\), we consider \(165^{\circ}\) as a sum of two known angles, such as \(165^{\circ} = 120^{\circ} + 45^{\circ}\). We can use the cosine of sum angle formula: \[\cos(a + b) = \cos a \cos b - \sin a \sin b\] where \(a = 120^{\circ}\) and \(b = 45^{\circ}\).

Step 2: Substitute known angle values and calculate cosine

Now, plug in the known exact trigonometric values: \[ \cos 120^{\circ} = -\frac{1}{2}, \quad \cos 45^{\circ} = \frac{\sqrt{2}}{2}, \quad \sin 120^{\circ} = \frac{\sqrt{3}}{2}, \quad \sin 45^{\circ} = \frac{\sqrt{2}}{2}\] Substituting these into the formula from Step 1 gives:\[\cos 165^{\circ} = \left(-\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right)\]

Step 3: Simplify the expression

Simplify the expression by performing the multiplications and then combining the terms:\[ \cos 165^{\circ} = -\frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} \]This can be combined into a single fraction:\[\cos 165^{\circ} = -\frac{\sqrt{2} + \sqrt{6}}{4}\]

Step 4: Finalize the exact value

The exact value of \(\cos 165^{\circ}\) is:\[-\frac{\sqrt{2} + \sqrt{6}}{4}\]}],

Key Concepts

Cosine Sum Formula

The cosine sum formula is incredibly useful when calculating the cosine of an angle that is the sum of two other angles. It is represented as follows:

• \[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]

This identity allows us to break down complex angles into combinations of simpler, known angles, making it perfect for angles like 165°, where the sum of 120° and 45° can be used.
First, it's important to select angles for which the trigonometric values are known and can be easily recalled or calculated, such as 30°, 45°, 60°, 90°, 120°, etc.
Once you've chosen your angles, just plug them into the formula as shown earlier to get to the cosine of the sum angle.

Exact Trigonometric Values

To solve problems using the cosine sum formula effectively, you need to know the exact trigonometric values of basic angles. These are special values that are typically memorized for angles like 0°, 30°, 45°, 60°, and 90°.
For example, here are some important values:

• \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \cos 120^{\circ} = -\frac{1}{2} \)
• \( \sin 120^{\circ} = \frac{\sqrt{3}}{2} \)

When we have these values, it simplifies the work in plugging into trigonometric identities like the cosine sum formula. Remember, knowing these exact values enables faster calculations and reduces the need for approximation, providing precise results.

Angle Addition in Trigonometry

The strategy of angle addition in trigonometry involves expressing a given angle as a sum or difference of two or more angles whose trigonometric values are more easily manageable.
For instance, the angle 165° might not have an immediately obvious cosine value, but it can be expressed as 120° + 45°. This makes it accessible through known trigonometric identities.
To implement this:

• Identify parts of the angle as sums or differences of known angles.
• Use appropriate trigonometric identities, such as cosine sum, sine sum, or tangent sum formulas.
• Plug in exact trigonometric values for each angle that you've broken down.

By utilizing angle addition, you can simplify complex problems into simpler steps, making it an efficient method for computing trigonometric values.