Problem 13 - Precalculus Mathematics for Calculus

Question

Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. $$\cos \frac{11 \pi}{12}$$

Step-by-Step Solution

Answer

The exact value of \( \cos \frac{11 \pi}{12} \) is \( -\frac{\sqrt{6} + \sqrt{2}}{4} \).

Step 1: Identify Parts of the Formula

In order to find the exact value of \( \cos \frac{11 \pi}{12} \), we can express this angle as the sum or difference of angles for which we know the cosine values. We'll use the identity: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \).

Step 2: Choose Suitable Angles

We can express \( \frac{11 \pi}{12} \) as \( \frac{3 \pi}{4} + \frac{\pi}{6} \) since \( \frac{3 \pi}{4} = \frac{9 \pi}{12} \) and \( \frac{\pi}{6} = \frac{2 \pi}{12} \). This sums up to \( \frac{11 \pi}{12} \).

Step 3: Apply the Addition Formula

Substitute \( a = \frac{3 \pi}{4} \) and \( b = \frac{\pi}{6} \) into the cosine addition formula: \[ \cos\left(\frac{3\pi}{4} + \frac{\pi}{6}\right) = \cos\frac{3\pi}{4} \cos\frac{\pi}{6} - \sin\frac{3\pi}{4} \sin\frac{\pi}{6} \]

Step 4: Substitute Known Values

We know \( \cos\frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \), \( \cos\frac{\pi}{6} = \frac{\sqrt{3}}{2} \), \( \sin\frac{3\pi}{4} = \frac{\sqrt{2}}{2} \), and \( \sin\frac{\pi}{6} = \frac{1}{2} \). Substitute these values into the equation: \[ \cos\frac{11\pi}{12} = \left(-\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right) \]

Step 5: Simplify the Expression

Carry out the multiplication and subtraction: \[ \cos\frac{11\pi}{12} = -\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = -\frac{\sqrt{6} + \sqrt{2}}{4} \]

Step 6: Final Result

So, the exact value of \( \cos\frac{11\pi}{12} \) is \( -\frac{\sqrt{6} + \sqrt{2}}{4} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are foundational in understanding how angles and their ratios relate to each other. These identities help simplify complex equations and make solving trigonometric problems more manageable. They are built upon the properties of sine, cosine, tangent, and other trigonometric functions.

Some of the key trigonometric identities include the Pythagorean identity, the angle sum and difference identities, and double angle identities. These identities let us rewrite expressions involving trigonometric functions into different forms, enabling us to find exact values for expressions involving angles not commonly found on the unit circle.

• Pythagorean Identity: \( \sin^2 a + \cos^2 a = 1 \)
• Angle Sum Identity: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)
• Angle Difference Identity: \( \sin(a - b) = \sin a \cos b - \cos a \sin b \)

These identities are incredibly useful when tackling problems that involve angles like \( \frac{11\pi}{12} \), as they help break down the problem into simpler components using known angles.

Cosine Addition Formula

The cosine addition formula is a specific type of trigonometric identity that allows us to find the cosine of the sum or difference of two angles. This formula is written as: \[ \cos(a + b) = \cos a \cos b - \sin a \sin b \].

Using this formula, we can express less common angles in terms of more familiar ones. For example, the angle \( \frac{11\pi}{12} \) can be broken down into a sum of angles like \( \frac{3\pi}{4} \) and \( \frac{\pi}{6} \), for which we know the cosine and sine values.

• Determine the angles \( a \) and \( b \) such that their sum gives the required angle.
• Use known trigonometric values for these angles to apply the formula.
• Simplify to reach the exact value of the expression.

By applying the cosine addition formula to \( \frac{11\pi}{12} \), we directly simplify the problem using angles \( \frac{3\pi}{4} \) and \( \frac{\pi}{6} \). This method capitalizes on known trigonometric values, making it much simpler to find exact solutions.

Exact Values in Trigonometry

Exact values in trigonometry refer to the precise values of trigonometric functions for certain standard angles. These angles are typically those found on the unit circle, such as \( 0, 30, 45, 60, \) and \( 90 \) degrees, or their radian equivalents. However, using trigonometric identities, we can find exact values for more complex angles as well.

For angles like \( \frac{11\pi}{12} \), trigonometric identities, including the cosine addition formula, allow us to express the angle as a sum or difference of standard angles. Then, we use known values to compute the trigonometric functions.

• Recognize standard angles and their trigonometric values.
• Use these values in identities to find exact results for complex expressions.
• Verify your solutions by ensuring the result is rational or involves common irrational numbers like \( \sqrt{2} \) or \( \sqrt{3} \).

By understanding and applying these exact values and identities, students become adept at navigating a variety of trigonometric problems.