Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find the exact value of \(\cos (A-B)\) for each given pair of values. \(A=180^{\circ}, B=30^{\circ}\)

Step-by-Step Solution

Answer

\( \cos(180^{\circ} - 30^{\circ}) = -\frac{\sqrt{3}}{2} \)

Step 1: Recall the Cosine Difference Formula

To find \( \cos(A-B) \), use the cosine difference formula: \( \cos(A-B) = \cos A \cos B + \sin A \sin B \).

Step 2: Identify Relevant Angles

Here, \( A = 180^{\circ} \) and \( B = 30^{\circ} \). Substitute these values into the formula for \( \cos(A-B) \).

Step 3: Evaluate \( \cos A \) and \( \sin A \)

For \( A = 180^{\circ} \), we have: \( \cos 180^{\circ} = -1 \) and \( \sin 180^{\circ} = 0 \).

Step 4: Evaluate \( \cos B \) and \( \sin B \)

For \( B = 30^{\circ} \), we have: \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \) and \( \sin 30^{\circ} = \frac{1}{2} \).

Step 5: Substitute Values into the Formula

Substitute the values found into the formula: \( \cos(A-B) = (-1) \left( \frac{\sqrt{3}}{2} \right) + (0) \left( \frac{1}{2} \right) \).

Step 6: Simplify the Expression

Simplify the expression: \( \cos(A-B) = -\frac{\sqrt{3}}{2} + 0 = -\frac{\sqrt{3}}{2} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are crucial in solving various problems in trigonometry. One such identity is the **Cosine Difference Formula** which helps in finding the cosine of the difference between two angles. This formula is given by \( \cos(A-B) = \cos A \cos B + \sin A \sin B \). It is a specific instance of angle formulas, which also include sum and tangent difference formulas. Understanding these identities allows us to transform complex trigonometric expressions into simpler, manageable forms.

• The cosine of the difference of two angles is directly related to the cosine and sine of the individual angles.
• These formulas are derived from the unit circle and fundamental properties of sine and cosine functions.
• Using these identities, you can evaluate trigonometric functions with non-standard angles.

By applying these formulas, we are able to find exact values of trigonometric functions that might otherwise require extensive calculations.

Angle Subtraction

Angle subtraction is an essential concept when working with trigonometric identities. It allows you to evaluate the trigonometric function of one angle minus another, such as in the difference formula mentioned above.
When dealing with angle subtraction:

• Identify the angles involved. Here, \( A = 180^{\circ} \) and \( B = 30^{\circ} \).
• Apply the difference formula accordingly, substituting the individual trigonometric values for sine and cosine.
• Use important angle results, such as knowing \( \cos 180^{\circ} \) and \( \sin 180^{\circ} \), to solve the problem.

This approach enables precise computation that relies on breaking down more complex angles into known components.
Angle subtraction techniques are not just limited to cosine but can also be applied to sine and tangent functions, making this tool versatile in trigonometry.

Exact Values of Trigonometric Functions

Finding the exact values of trigonometric functions involves using known angle properties and identities. When specific angle measures are involved, such as \( 180^{\circ} \) or \( 30^{\circ} \), it's crucial to understand their properties.

• For \( 180^{\circ} \), the exact values are: \( \cos 180^{\circ} = -1\) and \( \sin 180^{\circ} = 0 \).
• For \( 30^{\circ} \), the values are \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \) and \( \sin 30^{\circ} = \frac{1}{2} \).

Using these values, you can accurately compute results in trigonometric operations involving angle differences, like the cosine of \( 180^{\circ} - 30^{\circ} \).
These known values often come from memorizing specific angle values on the unit circle, enabling students to derive solutions quickly and accurately.
Understanding these exact values greatly simplifies the work that goes into evaluating expressions and enhances the problem-solving toolkit available to students.