Problem 17 - A Graphical Approach to Precalculus with Limits

Question

Use an identity to write each expression as a single trigonometric function or as a single number in exact form. Do not use a calculator. $$\cos ^{2} 15^{\circ}-\sin ^{2} 15^{\circ}$$

Step-by-Step Solution

Answer

\( \frac{\sqrt{3}}{2} \)

Step 1: Identify the Trigonometric Identity

The expression given is \( \cos^2 15^{\circ} - \sin^2 15^{\circ} \). Recognize that this is in the form of a well-known trigonometric identity: the difference of squares identity in trigonometry, which is \( \cos^2 \theta - \sin^2 \theta = \cos(2\theta) \).

Step 2: Apply the Identity

Using the trigonometric identity \( \cos^2 \theta - \sin^2 \theta = \cos(2\theta) \), substitute \( \theta = 15^{\circ} \) into the formula. This gives us \( \cos(2 \times 15^{\circ}) = \cos(30^{\circ}) \).

Step 3: Find the Exact Value

Identify the exact value of \( \cos(30^{\circ}) \). From known trigonometric values, \( \cos(30^{\circ}) = \frac{\sqrt{3}}{2} \). Therefore, the simplified form of the expression is \( \frac{\sqrt{3}}{2} \).

Key Concepts

Difference of Squares

The difference of squares is a powerful algebraic tool often used in simplifying trigonometric identities. It represents any expression in the form of \( a^2 - b^2 \) and can be simplistically represented by the identity \( a^2 - b^2 = (a + b)(a - b) \). In trigonometry, this takes on a special form, particularly with cosine and sine functions.

For example, \( \cos^2 \theta - \sin^2 \theta \) can be transformed using the identity \( \cos(2\theta) \). This allows us to easily simplify expressions that might initially appear complex.

Additionally, understanding how the difference of squares works allows for quick mental checks. It especially comes in handy when you need to apply further identities or find exact trigonometric values without resorting to a calculator.

Exact Trigonometric Values

Exact trigonometric values refer to the precise, often non-decimal forms of basic angles like \( 30^{\circ}, 45^{\circ}, \) and \( 60^{\circ} \). These values have been memorized by many students because they appear frequently in both theoretical and applied problems. Standard forms often include simple fractions and square roots.

For \( \cos(30^{\circ}) \), the exact trigonometric value is \( \frac{\sqrt{3}}{2} \), a value often encountered and essential in further computations. Recognizing these values quickly can significantly speed up solving processes in trigonometry.

Knowing exact values by heart not only aids in computations but also builds confidence in problem-solving. It facilitates checking work rapidly and ensures understanding of fundamental trigonometric properties and their applications.

Angle Simplification

Angle simplification is an important skill when dealing with trigonometric expressions. It involves reducing complex angle expressions into simpler terms using identities. This can involve angle doubling, halving, or even combining different angles to form compound angle identities.

In our specific case, recognizing \( \cos^2 15^{\circ} - \sin^2 15^{\circ} = \cos(30^{\circ}) \) is a matter of simplification through the double angle identity. This simply means applying the identity to see the angle \( 2 \times 15^{\circ} = 30^{\circ} \). By simplifying the angle, computations become straightforward and manageable.

Angle simplification not only aids in obtaining exact trigonometric values but also reduces potential errors in further calculations. Employing these strategies effectively can make complex trigonometric challenges more approachable and easier to solve.