Problem 12 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find the exact value of \(\cos (A-B)\) for each given pair of values. \(A=45^{\circ}, B=270^{\circ}\)

Step-by-Step Solution

Answer

The exact value of \( \cos(45^{\circ} - 270^{\circ}) \) is \(-\frac{\sqrt{2}}{2}\).

Step 1: Recall the Cosine Difference Identity

To find \( \cos(A-B) \), use the identity: \( \cos(A-B) = \cos A \cdot \cos B + \sin A \cdot \sin B \).

Step 2: Determine Cosine and Sine of Given Angles

For \( A = 45^{\circ} \), we have \( \cos A = \frac{\sqrt{2}}{2} \) and \( \sin A = \frac{\sqrt{2}}{2} \).For \( B = 270^{\circ} \), we have \( \cos B = 0 \) and \( \sin B = -1 \).

Step 3: Substitute Values into the Identity

Substitute the values found in Step 2 into \( \cos(A-B) = \cos A \cdot \cos B + \sin A \cdot \sin B \):\[ \cos(45^{\circ} - 270^{\circ}) = \left(\frac{\sqrt{2}}{2}\right) \cdot 0 + \left(\frac{\sqrt{2}}{2}\right) \cdot (-1) \]

Step 4: Simplify the Expression

Simplify the expression:\[ \cos(45^{\circ} - 270^{\circ}) = 0 - \frac{\sqrt{2}}{2} = -\frac{\sqrt{2}}{2} \]

Step 5: Final Step: Provide the Exact Value

Thus, the exact value of \( \cos(A-B) \) for \( A = 45^{\circ} \) and \( B = 270^{\circ} \) is \(-\frac{\sqrt{2}}{2}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical tools that relate the angles and lengths of a triangle. These identities are essential in solving a wide range of problems in trigonometry. One such useful identity is the Cosine Difference Identity. This identity helps find the cosine of the difference between two angles. It states:

• \[ \cos(A-B) = \cos A \cdot \cos B + \sin A \cdot \sin B\]

Using this identity, you can break down complex trigonometric expressions into simpler forms by using known cosine and sine values of individual angles. This method is particularly useful in situations like finding the exact value of cosine when two angles are subtracted. Understanding and applying trigonometric identities effectively simplifies many mathematical problems.

Cosine and Sine Values

Accurate determination of cosine and sine values for specific angles is vital in trigonometry. These values are fundamental in using trigonometric identities. Often, angles such as \(45^{\circ}\), \(90^{\circ}\), and \(270^{\circ}\) are encountered, where knowing their trigonometric function values can drastically simplify calculations.For instance:

• For \(A = 45^{\circ}\), the values are:
  \( \cos A = \frac{\sqrt{2}}{2} \)
  \( \sin A = \frac{\sqrt{2}}{2} \)
• For \(B = 270^{\circ}\), the values are:
  \( \cos B = 0 \)
  \( \sin B = -1 \)

These values derive from the unit circle, which is a circle with a radius of one centered at the origin of a coordinate plane. The unit circle simplifies the understanding of trigonometric functions because it provides a visual representation of their values at different angles. Knowing these values allows you to apply them in various trigonometric identities effectively.

Angle Subtraction

Angle subtraction is a critical calculation in trigonometry. When we are asked to find \(\cos(A-B)\), it means we want to determine the cosine of the difference between two angles, \(A\) and \(B\). Here, angle subtraction becomes crucial in using the Cosine Difference Identity, as it helps to find the combined effect of these angles.By applying the identity:

• \( \cos(45^{\circ} - 270^{\circ}) \)

We incorporate the precise cosine and sine values of each angle:

• \( \cos A = \frac{\sqrt{2}}{2}, \sin A = \frac{\sqrt{2}}{2} \)
• \( \cos B = 0, \sin B = -1 \)

Substituting these values into the identity, we simplify the subtraction:
\[ \cos(45^{\circ} - 270^{\circ}) = \left(\frac{\sqrt{2}}{2}\right) \cdot 0 + \left(\frac{\sqrt{2}}{2}\right) \cdot (-1) \]
Which results in:
\[ 0 - \frac{\sqrt{2}}{2} = -\frac{\sqrt{2}}{2} \]By understanding angle subtraction and applying trigonometric identities accurately, the exact value of complex expressions can be found efficiently.