Problem 7 - Precalculus

Question

In Exercises 7 - 21 , use the Sum and Difference Identities to find the exact value. You may have need of the Quotient, Reciprocal or Even / Odd Identities as well. $$ \cos \left(75^{\circ}\right) $$

Step-by-Step Solution

Answer

\(\cos(75^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4}\).

Step 1: Express 75° as a Sum

To use the sum identity, express \(75^\circ\) as a sum of angles with known trigonometric values. A common choice is \(75^\circ = 45^\circ + 30^\circ\).

Step 2: Apply the Cosine Sum Identity

The formula for the cosine of a sum is \(\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\). Using \(a = 45^\circ\) and \(b = 30^\circ\), we have \(\cos(75^\circ) = \cos(45^\circ)\cos(30^\circ) - \sin(45^\circ)\sin(30^\circ)\).

Step 3: Use Known Values for Trigonometric Functions

Recall the exact trigonometric values: \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\), \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\), and \(\sin(30^\circ) = \frac{1}{2}\).

Step 4: Substitute Values

Substitute the known values into the formula: \(\cos(75^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)\).

Step 5: Simplify the Expression

Compute the individual products: \(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{4}\) and \(\frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{4}\). Then, subtract: \(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}\).

Step 6: Final Answer

Thus, the exact value of \(\cos(75^\circ)\) is \(\frac{\sqrt{6} - \sqrt{2}}{4}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are formulas that involve the trigonometric functions which are constant for all angles. They are incredibly useful when solving problems involving angles because they allow us to transform and simplify expressions. These identities help in expressing trigonometric functions in different forms.

• Common trigonometric identities include Pythagorean, reciprocal, and even/odd identities.
• The Pythagorean identity states that for any angle \( \theta \), \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
• Reciprocal identities express functions like cosecant, secant, and cotangent as reciprocals of sine, cosine, and tangent respectively.
• Even and odd identities relate the function's value to its other angles, like \( \cos(-\theta) = \cos(\theta) \) and \( \sin(-\theta) = -\sin(\theta) \).

These identities are essential for simplifying expressions and solving problems, such as those involving sum and difference of angles.

Cosine Sum Identity

The cosine sum identity is a specific formula used to find the cosine of the sum of two angles. This identity is particularly useful when you need to calculate the cosine of angles that aren't among the common known angles, like 75 degrees.

• The identity is given by: \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \).
• Using this, the cosine of an angle expressible as a sum of known angles is easily computed.
• For instance, in our exercise, \( 75^{\circ} = 45^{\circ} + 30^{\circ} \). Using known values of \( \cos(45^{\circ}) \) and \( \cos(30^{\circ}) \), and the sine counterparts, allows us to find \( \cos(75^{\circ}) \).

This identity simplifies calculations and provides exact values for trigonometric expressions not immediately solvable by geometric interpretations alone.

Exact Trigonometric Values

Exact trigonometric values refer to values of trigonometric functions at specific angles that are well-defined and widely known. These angles include 0°, 30°, 45°, 60°, and 90°, among others. You don't need a calculator to find these values because they are memorized for quick reference.

• For example, \( \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \) and \( \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \).
• These values arise frequently in mathematical problems and are derived from the unit circle.
• Using exact values helps in performing accurate calculations when applying sum and difference identities.

Being familiar with these values allows for a smoother problem-solving process as they are integral in simplifying expressions and avoiding approximation errors.

Angle Sum Formulas

Angle sum formulas are invaluable tools in trigonometry, allowing the computation of trigonometric functions for sums and differences of angles. These formulas enable transformation of complex expressions into simpler, manageable ones.

• The primary formulas include:
  • \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \)
  • \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \)
  • \( \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \)
• These formulas facilitate easy evaluations of functions like \( \cos(75^{\circ}) \), using simpler angles.
• They are extensions of basic trigonometric principles that help in expressing functions combined with other operations.

By employing angle sum formulas, we not only solve problems more efficiently but also enhance our understanding of the intrinsic relationships between trigonometric functions.