Problem 66 - Precalculus with Limits

Question

In Exercises 65-68, use the co-function identities to evaluate the expression without using a calculator. \( \cos^2 55^\circ + \cos^2 35^\circ \)

Step-by-Step Solution

Answer

The value of \( \cos^2 {55^{\circ}} + \cos^2 {35^{\circ}} \) is 1.

Step 1: Identify complementary angles

Notice that the two angles, 55 degrees and 35 degrees, are complementary – they add up to 90 degrees.

Step 2: Identify co-function identity

We know that \( \cos^2{x} + \sin^2{x} = 1 \), from the pythagorean identity in trigonometry. We can also say that for any acute angle \( x \), we have \( \cos^2{x} + \cos^2{(90^{\circ}-x)} = 1 \), because \(\sin{x} = \cos{(90^{\circ}-x)} \). In our case, we don't know \( \cos {55^{\circ}} \) or \( \cos {35^{\circ}} \), but we do know that they are complementary, so we can use this alternative form of the identity.

Step 3: Apply the co-function identity to the given expression

Substitute 55 degrees for \( x \) in the formula from the previous step: \( \cos^2{55^{\circ}} + \cos^2{35^{\circ}} = 1 \).

Key Concepts

Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. This is important because it allows us to use certain trigonometric identities. If one angle is known, the complementary angle can easily be found by subtracting the known angle from 90 degrees. For example, if you have a 35-degree angle, its complement will be 55 degrees because 35 plus 55 equals 90. This relationship is useful in trigonometric calculations, particularly when dealing with co-function identities. Knowing that angles are complementary helps us establish certain trigonometric relationships without knowing individual angle values.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. They are handy tools in simplifying expressions and solving equations. A few well-known identities include:

• Pythagorean Identities
• Co-Function Identities
• Sum and Difference Identities

Each of these categories provides a different set of rules that help to analyze and simplify trigonometric expressions. Co-function identities are particularly useful for complementary angles. For instance, for any acute angle \(x\), the identities \(\sin{x} = \cos{(90^{\circ}-x)}\) and \(\cos{x} = \sin{(90^{\circ}-x)}\) hold. This comes in handy when comparing the trigonometric values of complementary angles.

Pythagorean Identity

The Pythagorean identity is one of the fundamental trigonometric identities derived from the Pythagorean theorem. It states that for any angle \(x\):

• \(\sin^2{x} + \cos^2{x} = 1\)

This identity implies that if you know one of the squares of either sine or cosine, you can easily determine the other.
The Pythagorean identity is not only a relationship between sine and cosine functions but also underpins many other trigonometric simplifications. It helps us understand that the sine and cosine values of an angle lie on a unit circle and form a right triangle where the sides squared plus the other side squared equals one.
In this exercise, using the co-function identity, we see a variant of this where, for complementary angles, the identity becomes \(\cos^2{x} + \cos^2{(90^{\circ}-x)} = 1\). This means if an angle and its complement are involved in an expression, the sum can simplify directly to 1—a result derived conveniently from this core identity.