Problem 179 - Precalculus

Question

On an interval of \([0,2 \pi),\) can the sine and cosine values of a radian measure ever be equal? If so, where?

Step-by-Step Solution

Answer

Yes, \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{5\pi}{4} \).

Step 1: Set Up Equation

We want to determine if there are any angles \( \theta \) in the interval \([0, 2\pi)\) where \( \sin(\theta) = \cos(\theta) \). To find this, we set the equation \( \sin(\theta) = \cos(\theta) \).

Step 2: Solve the Equation

To solve \( \sin(\theta) = \cos(\theta) \), divide both sides by \( \cos(\theta) \), assuming \( \cos(\theta) eq 0 \), to get \( \tan(\theta) = 1 \).

Step 3: Find Solutions for Tangent

The equation \( \tan(\theta) = 1 \) is true when \( \theta = \frac{\pi}{4} + n\pi \) for integers \( n \). However, we want \( \theta \) in the interval \([0, 2\pi)\).

Step 4: Determine Valid \( \theta \) in the Interval

Within the interval \([0, 2\pi)\), the solutions are \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{5\pi}{4} \). These are the angles where \( \tan(\theta) = 1 \).

Step 5: Verify Solutions

At \( \theta = \frac{\pi}{4} \): \( \sin(\theta) = \cos(\theta) = \frac{\sqrt{2}}{2} \). Similarly, at \( \theta = \frac{5\pi}{4} \), \( \sin(\theta) = \cos(\theta) = -\frac{\sqrt{2}}{2} \). Both values satisfy \( \sin(\theta) = \cos(\theta) \).

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry, often abbreviated as "sin". It relates to the ratio of the opposite side to the hypotenuse in a right-angled triangle. When you think about the unit circle, the sine of an angle is the y-coordinate of the point on the circle corresponding to that angle.

The sine function is periodic, which means it repeats its values in regular intervals. Specifically, the sine function has a period of \( 2\pi \), meaning that \( \sin(\theta) = \sin(\theta + 2\pi) \) for any angle \( \theta \). This property of periodicity is crucial when analyzing the behavior of the sine function over various intervals.

• The sine function starts at 0 when \( \theta = 0 \), reaches its maximum value of 1 when \( \theta = \frac{\pi}{2} \), decreases back to 0 at \( \theta = \pi \), continues to -1 at \( \theta = \frac{3\pi}{2} \), and returns to 0 at \( \theta = 2\pi \).
• This pattern repeats for every subsequent \( 2\pi \) interval.

Understanding the properties of the sine function helps solve trigonometric equations and understand how different functions relate to each other.

Cosine Function

The cosine function, abbreviated as "cos", is another pillar of trigonometry. Similar to the sine function, it also arises from the ratios in a right-angled triangle, but in this case, it is the ratio of the adjacent side to the hypotenuse. On the unit circle, the cosine of an angle is represented by the x-coordinate of the corresponding point on the circle.

The cosine function shares the same periodicity as the sine function, repeating every \( 2\pi \) interval, which makes it easy to analyze over any given range. This periodic nature means that \( \cos(\theta) = \cos(\theta + 2\pi) \), ensuring that the function's values are consistent every full circle around the unit circle.

• The cosine function reaches its maximum value of 1 at \( \theta = 0 \), decreases to 0 at \( \theta = \frac{\pi}{2} \), continues to a minimum of -1 at \( \theta = \pi \), goes back to 0 at \( \theta = \frac{3\pi}{2} \), and returns to 1 by the time \( \theta = 2\pi \).
• Because of its shape and pattern, it closely mirrors the behavior of the sine function, but is phase-shifted by \( \frac{\pi}{2} \).

The relationship between the sine and cosine functions is key to understanding trigonometric identities and solving complex trigonometric equations.

Tangent Function

The tangent function, abbreviated as "tan", is another essential trigonometric function. It is typically defined as the ratio of the sine function to the cosine function, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). In the context of the unit circle, the tangent can be seen as the slope of the line that passes through the origin and the point on the circle.

Unlike the sine and cosine functions, the tangent function has a shorter period, repeating its values every \( \pi \) rather than \( 2\pi \). This difference arises because the function \( \tan(\theta) \) becomes undefined at angles where \( \cos(\theta) = 0 \), notably at \( \theta = \frac{\pi}{2} + n\pi \), where \( n \) is an integer.

• The tangent function is positive in the first and third quadrants, where sine and cosine share the same sign, and negative in the second and fourth quadrants, where they differ.
• Key angles occur at \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{5\pi}{4} \) within the interval \([0, 2\pi)\), where \( \tan(\theta) = 1 \).

Understanding the tangent function, especially its periodicity and undefined points, is crucial for solving trigonometric equations involving tangent values.