Problem 10 - Precalculus with Limits

Question

In Exercises 7 - 12, find the exact value of each expression. (a) \( \cos\left(120^\circ + 45^\circ\right) \) (b) \( \cos 120^\circ + 45^\circ \)

Step-by-Step Solution

Answer

The exact values are: (a) - \( \frac{\sqrt{2}}{2} \) (b) 0

Step 1: Evaluate (a) \( \cos(120^\circ + 45^\circ) \)

Use the sum of angles formula for cosine, substituting \( a = 120^\circ \) and \( b = 45^\circ \). So \( \cos(120^\circ + 45^\circ) = \cos 120^\circ \cos 45^\circ + \sin 120^\circ \sin 45^\circ \). Evaluate each term using standard trigonometric values for these angles to find the result.

Step 2: Evaluate (b) \( \cos 120^\circ + \cos 45^\circ \)

Unlike part (a), this does not involve a trigonometric identity. Simply calculate the cosine of each angle and add the results together. Use standard trigonometric values for these angles.

Key Concepts

Angle Sum Formula

The angle sum formula is a fundamental identity in trigonometry. It is used to determine the cosine (or sine) of the sum of two angles. Specifically for cosine, the formula is:

• \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)
• However, a common form also used especially in textbooks is adding: \( \cos(a + b) = \cos a \cos b + \sin a \sin b \)

This formula allows us to break down complex angle expressions into simpler parts using known trigonometric values.
For instance, when we need to evaluate \( \cos(120^\circ + 45^\circ) \), we apply the angle sum formula as follows:

• Identify the individual angles: \( \cos 120^\circ \) and \( \cos 45^\circ \)
• Apply the formula: \( \cos 120^\circ \cos 45^\circ + \sin 120^\circ \sin 45^\circ \)

Once the sum formula is set up, each trigonometric function is replaced with its standard value, which simplifies the computation.
This method is efficient and crucial for solving complex trigonometric expressions in various math problems.

Cosine Function

The cosine function is a staple in trigonometry, often represented as \( \cos \).
It is a periodic function with a period of \(360^\circ\) or \(2\pi\) radians, meaning it repeats its values in regular intervals. Here are several key points about the cosine function:

• The range is \([-1, 1]\), indicating that the cosine of any angle cannot exceed these bounds.
• It is an even function, which means \( \cos(-x) = \cos(x) \).
• It starts from \(1\) at \(0^\circ\), decreases to \(-1\) at \(180^\circ\), and returns to \(1\) at \(360^\circ\).

These properties help in various applications, from calculating phase in physics to determining sound waveforms.
In trigonometry problems, the cosine function often interacts with angle sum formulas, as seen in the given exercise.
The sum of angles expression \( \cos(120^\circ + 45^\circ) \) calculates the cosine of a compound angle using this consistent pattern the cosine function follows.

Standard Trigonometric Values

Standard trigonometric values refer to cosine and sine values for commonly encountered angles like \(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ,\) and so forth.
These values are often memorized due to their frequency in mathematical problems. For example:

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

These standard values simplify the computation within trigonometric identities like the angle sum formula.
Rather than calculating from scratch, we can plug in these values directly into equations.
In exercises, falling back on standard values helps quickly solve expressions and confirm the accuracy of solutions.
It's crucial to have these trigonometric values at your fingertips, as they are foundational for solving problems efficiently.