Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find the exact value of \(\cos (A-B)\) for each given pair of values. \(A=90^{\circ}, B=60^{\circ}\)

Step-by-Step Solution

Answer

The exact value of \( \cos(90^{\circ} - 60^{\circ}) \) is \( \frac{\sqrt{3}}{2} \).

Step 1: Recall the formula for cosine of angle difference

The formula for the cosine of difference of two angles is: \( \cos(A-B) = \cos A \cos B + \sin A \sin B \). This formula helps to calculate the cosine of the difference between two given angles.

Step 2: Input the given angle values

Substitute the given values \( A = 90^{\circ} \) and \( B = 60^{\circ} \) into the formula: \[ \cos(90^{\circ} - 60^{\circ}) = \cos 90^{\circ} \cos 60^{\circ} + \sin 90^{\circ} \sin 60^{\circ}. \]

Step 3: Find trigonometric values of standard angles

Use the known trigonometric values: \( \cos 90^{\circ} = 0 \), \( \sin 90^{\circ} = 1 \), \( \cos 60^{\circ} = \frac{1}{2} \), and \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \).

Step 4: Substitute trigonometric values and simplify

Replace the trigonometric values in the equation from Step 2: \[ \cos(90^{\circ} - 60^{\circ}) = (0)(\frac{1}{2}) + (1)(\frac{\sqrt{3}}{2}) = \frac{\sqrt{3}}{2}. \] Therefore, the exact value of \( \cos(90^{\circ} - 60^{\circ}) \) is \( \frac{\sqrt{3}}{2} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are true for every value of the variables involved. These identities play a crucial role in simplifying complex trigonometric expressions and solving equations. One of the essential trigonometric identities used in problems involving angle differences is the cosine of angle difference formula. This identity is expressed as:

• \( \cos(A-B) = \cos A \cos B + \sin A \sin B \)

This formula provides us with a way to calculate the cosine of the difference between two angles using the sine and cosine of each individual angle. Understanding and memorizing these identities can greatly aid students in trigonometry, as they frequently appear in multiple mathematical contexts. Remember also that these identities help to establish relationships among the trigonometric functions, such as sine, cosine, etc., which are pivotal in solving angular problems across geometry and physics.

Standard Angle Values

Standard angle values refer to specific angles—usually expressed in degrees or radians—whose trigonometric function values are commonly known and used. Angles like \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), and \(90^\circ\) are examples of standard angles. For example, the following are the trigonometric values for some common standard angles:

• \( \cos 0^\circ = 1 \)
• \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)
• \( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
• \( \cos 60^\circ = \frac{1}{2} \)
• \( \cos 90^\circ = 0 \)
• \( \sin 90^\circ = 1 \), \( \sin 60^\circ = \frac{\sqrt{3}}{2} \), and so on.

Knowing these values makes it easy to solve problems like the cosine angle difference. Using the standard angle values, we can solve expressions faster without a calculator, and develop a deeper comprehension of trigonometric functions.

Exact Trigonometric Values

Exact trigonometric values are those that can be expressed in terms of square roots, fractions, or whole numbers, rather than decimals. These values give us a precise and accurate way to work with trigonometric expressions, something particularly important in many fields such as engineering and science. For example, when we calculated \(\cos(90^{\circ} - 60^{\circ})\), we ended up with \(\frac{\sqrt{3}}{2}\), which is an exact trigonometric value.When dealing with trigonometric expressions, exact values are often preferred over their decimal equivalents because:

• They offer precision and prevent rounding errors.
• They facilitate further mathematical manipulation where simplification is needed.
• They align with symbolic computation often used in advanced mathematics.

Being comfortable with finding and manipulating exact trigonometric values sharpens critical thinking and problem-solving skills, providing a great advantage in various mathematical applications.