Problem 3 - Precalculus Essentials

Question

Find the exact value of each expression.. $$\cos \left(\frac{3 \pi}{4}-\frac{\pi}{6}\right)$$

Step-by-Step Solution

Answer

The exact value of \(\cos \left(\frac{3 \pi}{4}-\frac{\pi}{6}\right)\) is \(-\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}\)

Step 1: Identify A and B

By examining the expression, it is found that A equals \(\frac{3 \pi}{4}\) and B equals \(\frac{\pi}{6}\).

Step 2: Compute the Cosine and Sine Values

The values for cosine and sine are determined. This gives us \(\cos\frac{3 \pi}{4} = -\frac{\sqrt{2}}{2}\), \(\sin\frac{3 \pi}{4} = \frac{\sqrt{2}}{2}\), \(\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}\) and \(\sin\frac{\pi}{6} =\frac{1}{2}\).

Step 3: Substitute into the Formula

The values of \(\cos A, \sin A, \cos B, \sin B\) are substituted into the formula \(\cos(A - B) = \cos A \cdot \cos B + \sin A \cdot \sin B\). So, \(\cos\left(\frac{3 \pi}{4}-\frac{\pi}{6}\right) = -\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}\).

Step 4: Simplify the Expression

The expression simplifies to \(-\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}\). Combining both terms leads to the final result.

Key Concepts

Cosine Difference Formula

Understanding the Cosine Difference Formula is essential when working with problems related to angle subtraction in trigonometry. The formula is expressed as:\[ \cos(A - B) = \cos A \cdot \cos B + \sin A \cdot \sin B \]This formula allows us to break down the cosine of a difference of two angles into a combination of products involving the cosines and sines of the individual angles. It is particularly useful because it transforms a complex problem into a step-by-step computation that involves simpler operations like multiplication and addition.In the given problem, we apply this formula by identifying two angles, \( A \) and \( B \), which are specifically given as fractions of \( \pi \), a common angle measure in trigonometry:

• \( A = \frac{3\pi}{4} \)
• \( B = \frac{\pi}{6} \)

This step is crucial for simplifying the expression accurately.When using the formula, ensure you have the correct values for both \( \cos \) and \( \sin \) of each angle, or the final result might be inaccurate.

Exact Values of Trigonometric Functions

Trigonometric functions often have special angles that yield exact values without the need for a calculator. Knowing these exact values helps in efficiently solving trigonometric equations and expressions.For angles such as \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), \( \frac{\pi}{2} \), and so on, the exact values can be memorized:

• \( \cos\frac{\pi}{6} = \frac{\sqrt{3}}{2} \)
• \( \sin\frac{\pi}{6} = \frac{1}{2} \)
• \( \cos\frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \)
• \( \sin\frac{3\pi}{4} = \frac{\sqrt{2}}{2} \)

These identifications are important, as they allow us to insert the exact values directly into trigonometric formulas, such as the one used in the earlier section. Not only does this ensure precision, but it also simplifies calculations by reducing reliance on decimal approximations, which can lead to errors in trigonometric problems.

Angle Subtraction in Trigonometry

Angle subtraction in trigonometry often requires specific formulas that facilitate numerical simplification. The significance of angle subtraction lies in its ability to express complex trigonometric expressions in terms that are easier to handle and calculate. By subtracting angles, such as \( \frac{3\pi}{4} - \frac{\pi}{6} \), we are effectively finding a new angle measure that can be broken down as a difference between known, *standard angles*.These measures are frequently seen in trigonometry, and understanding the transformations that take place enhances comprehension and efficiency:

• Converts difficult angle measures into sums or differences of known angles.
• Makes use of trigonometric identities to solve expressions.
• Utilizes exact values for simplification and clarity.

To solve the original expression, after defining \( A \) and \( B \), one can substitute into the cosine difference formula, making the problem manageable and allowing for the quick finding of solutions, ultimately simplifying the expression into an exact value.