Problem 9 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find the exact value of \(\cos (A-B)\) for each given pair of values. \(A=30^{\circ}, B=90^{\circ}\)

Step-by-Step Solution

Answer

The exact value of \(\cos(30^\circ - 90^\circ)\) is \(\frac{1}{2}\).

Step 1: Understand the Formula

To find \(\cos(A-B)\), we use the cosine difference identity: \[\cos(A-B) = \cos A \cos B + \sin A \sin B.\] We need to find \(\cos(30^\circ)\), \(\cos(90^\circ)\), \(\sin(30^\circ)\), and \(\sin(90^\circ)\).

Step 2: Calculate Trigonometric Values of A

\(A = 30^\circ\). The cosine and sine values are as follows:\[\cos(30^\circ) = \frac{\sqrt{3}}{2}, \quad \sin(30^\circ) = \frac{1}{2}.\]

Step 3: Calculate Trigonometric Values of B

\(B = 90^\circ\). The cosine and sine values are as follows:\[\cos(90^\circ) = 0, \quad \sin(90^\circ) = 1.\]

Step 4: Substitute Values into Formula

Substitute the trigonometric values into the formula:\[\cos(A-B) = \cos(30^\circ)\cos(90^\circ) + \sin(30^\circ)\sin(90^\circ).\]This becomes:\[\cos(A-B) = \left(\frac{\sqrt{3}}{2}\right)(0) + \left(\frac{1}{2}\right)(1).\]

Step 5: Simplify the Expression

Simplify the expression:\[\cos(A-B) = 0 + \frac{1}{2} = \frac{1}{2}.\] Thus, \(\cos(30^\circ - 90^\circ) = \frac{1}{2}.\)

Key Concepts

Understanding Trigonometric Identities

Trigonometric identities are fundamental tools in trigonometry that reveal relationships between different trigonometric functions. These identities help simplify complex trigonometric problems and allow us to find exact values for certain expressions. The cosine difference identity, which is used in this exercise, is one of these crucial relationships. It is given by:\[ \cos(A-B) = \cos A \cos B + \sin A \sin B. \]This identity helps us express the cosine of a difference of two angles in terms of the cosines and sines of the individual angles. By utilizing trigonometric identities, we can solve problems more efficiently and understand the interconnections between trigonometric functions. This is particularly useful when we want to find exact trigonometric values for non-standard angles.

Using Angle Difference Formulas

The angle difference formulas, such as the cosine difference formula, are special trigonometric identities used to calculate the trigonometric values of an angle expressed as a difference of two angles. In this case, the formula helps us find \( \cos(A-B) \) when given angles \( A \) and \( B \). For the problem \( A = 30^{\circ} \) and \( B = 90^{\circ} \), we use the cosine difference identity to break down and simplify the expression. By plugging in the known trigonometric values, we simplify the problem:- \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \)- \( \sin(30^{\circ}) = \frac{1}{2} \)- \( \cos(90^{\circ}) = 0 \)- \( \sin(90^{\circ}) = 1 \)This step-by-step substitution and simplification allow us to find the exact value of \( \cos(30^{\circ} - 90^{\circ}) \) straightforwardly.

Exploring Exact Trigonometric Values

Exact trigonometric values are values of trigonometric functions that are known precisely rather than being approximated. These values are often memorized for specific angles such as \(0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, \) and \(90^{\circ}\). Learning these values is crucial because they are used frequently in solving trigonometric problems.For example:

• \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2} \)
• \(\sin(30^{\circ}) = \frac{1}{2} \)
• \(\cos(90^{\circ}) = 0 \)
• \(\sin(90^{\circ}) = 1 \)

These exact values allow us to calculate things like \(\cos(A-B)\), or any other trigonometric expressions, precisely without a calculator. Understanding these exact values aids in mental calculations and provides deeper insight into the behavior of trigonometric functions at key angles.