Problem 69 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the cofunction identities to evaluate the expression without using a calculator. $$\cos ^{2} 20^{\circ}+\cos ^{2} 52^{\circ}+\cos ^{2} 38^{\circ}+\cos ^{2} 70^{\circ}$$

Step-by-Step Solution

Answer

The value of the expression \( cos^2 20° + cos^2 52° + cos^2 38° + cos^2 70° = 4 \)

Step 1: Understand how to use the cofunction identities

Remember the cofunction identity for cosine, which is \(cos(A) = sin(90° - A)\). This means the cosine of any angle is equal to the sine of its complementary angle.

Step 2: Apply the cofunction identity to the expression

Knowing that 20° and 70° are complements, as well as 52° and 38°, use the cofunction identities: \(cos^2 20° = sin^2 70°\), \(cos^2 70° = sin^2 20°\), \(cos^2 52° = sin^2 38°\), and \(cos^2 38° = sin^2 52°\). Replace these in the original problem.

Step 3: Add and Simplify

Add the equations: \(sin^2 70° + sin^2 20° + sin^2 52° + sin^2 38°\). Remember, \( sin^2 A + cos^2 A = 1\), here every angle A and its complement add up to 1. So, \( sin^2 70° + cos^2 70° = 1 \) and \( sin^2 20° + cos^2 20° = 1 \). Therefore, the sum of all the terms is 4.

Key Concepts

Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. This is a fundamental concept in trigonometry, particularly when dealing with cofunction identities. When two angles are complementary, we can use this relationship to find the trigonometric functions of one angle using the functions of the other.
For example, if we have two angles, such as 20° and 70°, they are complementary because 20° + 70° equals 90°. This relationship is useful because it helps simplify the computation of trigonometric functions by employing known values of their complements. Understanding how complementary angles work is crucial for using cofunction identities effectively.

Cosine Function

The cosine function is one of the primary trigonometric functions and is essential in the study of right-angle triangles and the unit circle. It is denoted as cos(θ), where θ represents the angle. The cosine function gives the ratio of the adjacent side to the hypotenuse in a right triangle.
In the context of the cosine function and complementary angles, we use the identity:

• \[\cos(A) = \sin(90° - A) \]

This identity shows how the cosine of an angle is related to the sine of its complementary angle. By utilizing this relationship, we simplify expressions and solve problems more easily, as shown in the original exercise.

Sine Function

The sine function, another core trigonometric function, is paired closely with the cosine function. It is written as sin(θ) and represents the ratio of the opposite side to the hypotenuse in a right-angled triangle.
With complementary angles, the sine function follows a key identity similar to the cosine function. This identity is represented as:

• \[\sin(A) = \cos(90° - A) \]

Thus, the sine of an angle is equal to the cosine of its complementary angle. Through this understanding, we establish valuable connections in solving trigonometric problems, like those presented in the exercise, where cosine squared terms are converted to sine squared terms using cofunction identities.

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables involved, given certain conditions. They allow us to simplify complex expressions and solve equations in trigonometry.
Among these identities, cofunction identities play a vital role when dealing with complementary angles. They relate the sine and cosine functions, such as:

• \[\sin^2 A + \cos^2 A = 1\]

This identity states that the sum of the square of the sine and the square of the cosine of the same angle equals 1, forming the foundation of many trigonometric simplifications. When working with exercises, understanding and applying these identities facilitate problem-solving, as each pair of sine and cosine terms for complementary angles simplifies the problem significantly.