Problem 68 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the cofunction identities to evaluate the expression without using a calculator. $$\cos ^{2} 18^{\circ}+\cos ^{2} 72^{\circ}$$

Step-by-Step Solution

Answer

The value of \(\cos ^{2} 18^{\circ}+\cos ^{2} 72^{\circ}\) is 1

Step 1: Apply the cofunction identity

The cofunction identity states that \(\cos(90^{\circ}-A) = \sin A\) or \(\sin(90^{\circ}-A) = \cos A\). Applying this to \(\cos ^{2} 72^{\circ}\), it becomes \(\cos ^{2} 72^{\circ} = \sin^{2}(90^{\circ} - 72^{\circ}) = \sin^{2} 18^{\circ}\). Hence the original expression becomes \(\cos ^{2} 18^{\circ}+\cos ^{2} 72^{\circ} = \cos ^{2} 18^{\circ}+\sin ^{2} 18^{\circ}\)

Step 2: Use the Pythagorean identity

The Pythagorean identity states that \(\cos^{2}A + \sin^{2}A = 1\), for any angle A. Hence \( \cos ^{2} 18^{\circ}+\sin ^{2} 18^{\circ} = 1 \)

Step 3: Final Result

So, the value of \(\cos ^{2} 18^{\circ}+\cos ^{2} 72^{\circ}\) is 1

Key Concepts

Understanding Cofunction Identities

Cofunction identities are a fundamental part of trigonometry, making it much easier to work with angles. These identities link complementary angles (angles that sum up to 90°). For instance, the cofunction identity for cosine is given by \( \cos(90^{\circ}-A) = \sin A \). This means the cosine of an angle can be expressed as the sine of its complementary angle.
In the context of our exercise, we use this identity to transform \( \cos^{2}72^{\circ} \) into \( \sin^{2}18^{\circ} \). This is done by recognizing that \( 72^{\circ} \) and \( 18^{\circ} \) are complementary angles. By applying the identity, \( \cos(90^{\circ} - 72^{\circ}) \) is changed into \( \sin 18^{\circ} \). This transformation is critical as it allows us to express the expression in terms of a single angle, simplifying calculations.

Exploring the Pythagorean Identity

The Pythagorean identity is one of the key identities in trigonometry, often used to simplify expressions. It states that for any angle \( A \), the equation \( \cos^{2}A + \sin^{2}A = 1 \) holds true. This identity arises from the Pythagorean theorem applied to a unit circle, where the hypotenuse is 1.
In our exercise, after applying the cofunction identity, the expression becomes \( \cos^{2}18^{\circ} + \sin^{2}18^{\circ} \). By utilizing the Pythagorean identity, we immediately recognize this is equal to 1. This demonstrates how powerful these identities are, as they simplify complex trigonometric expressions into numerical values quickly.
Let's summarize:

• This identity helps evaluate trigonometric expressions by converting them into easily calculable forms.
• It confirms that if you know one function value, the other can be determined.

Role of Angle Evaluation in Trigonometry

Evaluating angles efficiently is essential in trigonometry. Understanding how to evaluate and manipulate angles plays a huge role in simplifying trigonometric expressions, like the one in our exercise. Angle evaluation often involves recognizing special angle relationships like complementary, supplementary, or when angles form parts of geometric figures.
By examining \(72^{\circ}\) and \(18^{\circ}\), the relationship becomes clear that these are complementary angles. Recognizing such patterns can be crucial as it allows the application of identities like the cofunction identity, which we used in this problem.
Effective angle evaluation also allows for efficient decision-making about which identities to apply, leading to faster and more accurate solutions in trigonometry. This skill can be honed through practice and familiarity with angles associated with frequently used identities. Remember:

• Always check if angles are related (like complementary or supplementary).
• Leverage known identity relationships to simplify calculations.