Problem 67 - Precalculus with Limits

Question

In Exercises 65-68, use the co-function 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 \(\cos^2 20^\circ + \cos^2 52^\circ + \cos^2 38^\circ + \cos^2 70^\circ\) is 2

Step 1: Identify correct pairs

Pair the terms in such a manner that the angles in each pair sum up to 90 degrees. This is possible in the given expression because \(20^\circ + 70^\circ = 90^\circ\) and \(52^\circ + 38^\circ = 90^\circ\). So, the pairs are \(\cos^2 20^\circ + \cos^2 70^\circ \) and \(\cos^2 52^\circ + \cos^2 38^\circ \)

Step 2: Use the co-function identities

The co-function identity for cosine is: \(\cos(90^\circ - \theta) = \sin(\theta)\). Therefore, \(\cos^2 20^\circ + \cos^2 70^\circ = 1\) and \(\cos^2 52^\circ + \cos^2 38^\circ = 1\). This is because the sum of the square of sine and cosine of the same angle is always 1.

Step 3: Evaluate the expression

Now, simply sum up the results from Step 2. The expression \(\cos^2 20^\circ + \cos^2 52^\circ + \cos^2 38^\circ + \cos^2 70^\circ = 1 + 1 = 2\)

Key Concepts

Co-function Identities

In trigonometry, co-function identities relate the ratios of complementary angles. Complementary angles are two angles that add up to 90 degrees. These identities are useful because they allow us to express trigonometric functions in terms of one another. For example, the co-function identity for the cosine function is expressed as \( \cos(90^\circ - \theta) = \sin(\theta) \). This means that the cosine of an angle is equal to the sine of its complementary angle.
This identity can help solve trigonometric problems by simplifying expressions. In our exercise, each pair of angles—\( 20^\circ \) and \( 70^\circ \), \( 52^\circ \) and \( 38^\circ \)—are complementary. This insight enables us to utilize the identities and establish that \( \cos^2 20^\circ + \cos^2 70^\circ = 1 \) and \( \cos^2 52^\circ + \cos^2 38^\circ = 1 \).
Understanding co-function identities not only helps simplify calculations but also reinforces the symmetrical properties of the trigonometric functions.

Cosine Function

The cosine function is a fundamental trigonometric function that measures the adjacent side over the hypotenuse in a right triangle. It describes the horizontal component of an angle in the unit circle.
The expression \( \cos \theta \) relates the angle \( \theta \) with the length of the side adjacent to \( \theta \) multiplied by the hypotenuse's length. Cosine is a periodic function, meaning it repeats its values in regular intervals. The periodicity of cosine is 360 degrees or \( 2\pi \) radians.
Cosine values are crucial in various domains like physics, engineering, and computer graphics because they help model oscillating phenomena, such as waves or circular motion. In problems like our exercise, it is important to understand how cosine pairs with other trigonometric ratios, particularly through co-function identities, for effective problem-solving.

Degrees in Trigonometry

Degrees are a way of measuring angles. One complete circle is 360 degrees. This measurement system is commonly used in trigonometry to facilitate the description of angle sizes without having to delve into the more complex radian measure.
In our exercise, the given angles are in degrees, which makes it easier to use the co-function identities because complementary angles straightforwardly add up to 90 degrees in this system. This clear relation between angles simplifies the application of trigonometric identities.
Learning to convert between degrees and radians is essential but sticking with degrees often makes initial learning simpler. Degrees provide an intuitive understanding of angle measure, reinforcing that two angles summing up to 90 degrees are complementary and plugging easily into co-function identities.