Problem 11 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find the exact value of \(\cos (A-B)\) for each given pair of values. \(A=60^{\circ}, B=270^{\circ}\)

Step-by-Step Solution

Answer

The exact value of \( \cos(60^{\circ} - 270^{\circ}) \) is \(-\frac{\sqrt{3}}{2}\).

Step 1: Write Down the Formula

The formula for the cosine of the difference between two angles is \( \cos(A-B) = \cos A \cos B + \sin A \sin B \). We will use this formula to find \( \cos(60^{\circ} - 270^{\circ}) \).

Step 2: Substitute the Given Angles

Substitute \( A = 60^{\circ} \) and \( B = 270^{\circ} \) into the formula. We want to find \( \cos(60^{\circ} - 270^{\circ}) = \cos(60^{\circ}) \cos(270^{\circ}) + \sin(60^{\circ}) \sin(270^{\circ}) \).

Step 3: Find the Trigonometric Values

Recall that \( \cos(60^{\circ}) = \frac{1}{2} \), \( \cos(270^{\circ}) = 0 \), \( \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \), and \( \sin(270^{\circ}) = -1 \).

Step 4: Plug in Values and Simplify

Substitute the trigonometric values into the formula: \( \cos(60^{\circ} - 270^{\circ}) = \left(\frac{1}{2}\right)(0) + \left(\frac{\sqrt{3}}{2}\right)(-1) \). This becomes \( 0 - \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2} \).

Key Concepts

Cosine Difference Identity

The cosine difference identity is a fundamental trigonometric formula that helps us find the cosine of the difference between two angles. This formula is quite useful in various mathematical calculations and can be used to simplify complex trigonometric expressions. The identity is given by:\[\cos(A-B) = \cos A \cos B + \sin A \sin B\]Here’s how it works:

• It breaks down the cosine of a difference into products of sines and cosines of the individual angles.
• This identity is derived from the unit circle and is used extensively in solving trigonometric equations and identities.
• It also allows for the conversion of non-standard angles into easily manageable forms.

This identity is especially handy when dealing with angles that are special angles, such as 30°, 45°, 60°, 90°, etc., where we can easily recall their exact trigonometric values to calculate the result.

Exact Trigonometric Values

To efficiently utilize trigonometric identities, it is crucial to know the exact trigonometric values of key angles such as 0°, 30°, 45°, 60°, 90°, and their multiples. These values allow us to bypass complex calculations and directly find results. Some commonly used exact values include:

• For 60°, \( \cos(60°) = \frac{1}{2} \) and \( \sin(60°) = \frac{\sqrt{3}}{2} \).
• For 270°, \( \cos(270°) = 0 \) and \( \sin(270°) = -1 \).

Understanding these values provides a quick shortcut for computation and is incredibly useful in applying trigonometric identities. By memorizing these values, solving problems involving specific angle measures becomes almost second nature, making it easier to focus on the steps of identities without getting bogged down by calculations.

Angle Subtraction in Trigonometry

Angle subtraction is a cornerstone in trigonometry that involves calculating the trigonometric functions of the difference between two angles. This concept uses identities, like the cosine difference identity, to simplify expressions:

• It requires one to mentally visualize the unit circle and understand the position of angles, especially when they span more than one quadrant.
• The operation typically revolves around transposing larger angles into related acute angles through subtractive methods.

For instance, in the exercise given with angles 60° and 270°, subtracting directly \( 60° - 270° \) presents us with \(-210°\), or equivalently, \( 150° \) when considering the angle in standard position. Recognizing the relationships and positions of angles allows us to use identities more effectively and solve problems with greater precision and speed.