Problem 97 - Precalculus : Building Concepts and Connections

Question

Find an angle s such that \(s \neq t, 0 \leq s<2 \pi\) and \(\cos s=\cos t\) $$t=\frac{\pi}{4}$$

Step-by-Step Solution

Answer

So, the angle \(s\) which satisfies the given conditions is \(s = \frac{7\pi}{4}\)

Step 1: Compute the value of cosine of given angle

Given that \(t = \frac{\pi}{4}\), compute the value of \(\cos t\), which is \(\cos(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}\)

Step 2: Apply the property of cosine

Since \(\cos(x)\) is an even function, \(\cos(-x) = \cos(x)\). Apply this property and equate \(\cos(-t)\) to \(\cos(s)\), thus \(\cos(-t) = \cos(s)\)

Step 3: Find the value of s

Replace \(t\) with \(\frac{\pi}{4}\) in the equation \(\cos(-t) = \cos(s)\), it then becomes \(\cos(-\frac{\pi}{4}) = \cos(s)\) . Hence, \(s = -t = -\frac{\pi}{4}\)

Step 4: Adjust s to lie in the correct quadrant

However, we are given that \(0 \leq s <2 \pi\) and \(s \neq t\), so we need to adjust \(s\) into the correct range. Add \(2\pi\) to \(s\)

Step 5: Final computation

Compute \(s + 2\pi\) which results in \(s = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4}\)

Key Concepts

Even Function

The cosine function, denoted as \( \text{cos} \), is known for its significant property of being an even function. An even function is defined by the symmetry in its graph with respect to the y-axis, meaning that for every input \( x \), the function's value at \( x \) is the same as at \( -x \). In mathematical terms, this is represented as \( f(x) = f(-x) \).

For instance, if we take the cosine of an angle \( t \), and its opposite \( -t \), their cosine values will be equal, \( \text{cos}(t) = \text{cos}(-t) \). This property is fundamental when solving trigonometric equations, as it allows us to find equivalent angles that produce the same cosine value but may lie in different quadrants of the unit circle.

When tasked to find an angle \( s \) such that \( s eq t \) and \( \text{cos}(s) = \text{cos}(t) \), remembering that cosine is an even function helps us to consider not only the angle given but also its additive inverse. This approach expands our options for solutions that satisfy the given conditions without altering the function's value.

Trigonometric Identities

Trigonometric identities are various equalities involving trigonometric functions that are true for all values within the functions' domains. They are immensely useful in simplifying and solving trigonometric equations. One of the commonly used basic trigonometric identities is the Pythagorean identity, which states that for any angle \( x \), \( \text{cos}^2(x) + \text{sin}^2(x) = 1 \).

Another important aspect of trigonometric identities is understanding the relationship between the trigonometric functions and the angles they represent. For example, knowing that \( \text{cos}(0) = 1 \) and \( \text{cos}(\frac{\text{π}}{2}) = 0 \) can aid in sketching the general shape of the cosine function or in determining specific function values without a calculator.

When solving a problem involving the cosine function, trigonometric identities like the symmetrical properties (such as the even function property of cosine) provide a systematic approach to finding all potential angles that yield the same value.

Radian Measure

Radian measure is a way of expressing angles by the length of the arc that they subtend on the unit circle, where the radius of the circle is one. One full revolution around the circle corresponds to \( 2\text{π} \) radians. Radians provide a more natural way of describing angles in the context of trigonometric functions compared to degrees.

In many mathematical problems, including those involving trigonometric functions, angles are typically expressed in radians. This is because radian measure often simplifies the mathematics involved, particularly when dealing with trigonometric identities and calculus.

For the given problem, angles are expressed in radians, with \( t = \frac{\text{π}}{4} \). When finding an angle \( s \) that corresponds to the same cosine value, using radian measure allows us to easily manipulate the angle to fit within the specified range of \( 0 \text{ inclusively to } 2\text{π} \text{ exclusively} \), ensuring we arrive at a solution that is consistent with the conditions presented.