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Chapter 8: Problem 41

Give a general formula for all the solutions List six solutions. \(\cos \theta=0\)

Short Answer

Expert verified
The general formula is \( \theta = \frac{\pi}{2} + k \pi \, \text{where k is an integer}\). Six solutions are \( \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, -\frac{\pi}{2}, -\frac{3\pi}{2}, -\frac{5\pi}{2} \).

Step by step solution

01

Understand the Basic Property of Cosine Function

The cosine function, \(\text{cos} \theta\), gives zero values at specific points on the unit circle. These points are where the angle \(\theta\) corresponds to the x-coordinate being zero.
02

Identify the Angles Where \( \text{cos} \theta = 0 \)

The cosine function is zero at \( \text{\theta} = \frac{\pi}{2} + k\pi\), where \( k \) is any integer. This is because the cosine graph intersects the \( \text{\theta} \) axis at these points repeatedly.
03

Write the General Formula

The general formula for all the solutions to the equation \( \text{cos} \theta = 0 \) is : \( \theta = \frac{\pi}{2} + k\pi \), where \( k \) is an integer.
04

List Six Specific Solutions

Using the general formula \( \theta = \frac{\pi}{2} + k\pi \), list the first six solutions by substituting different integer values for \( k \): \( k = 0 : \frac{\pi}{2} \, \ k = 1 : \frac{3\pi}{2} \, \ k = 2 : \frac{5\pi}{2} \, \ k = -1 : -\frac{\pi}{2} \, \ k = -2 : -\frac{3\pi}{2} \, \ k = -3 : -\frac{5\pi}{2} \). Hence, the six solutions are \( \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, -\frac{\pi}{2}, -\frac{3\pi}{2}, -\frac{5\pi}{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

cosine function
The cosine function, often denoted as \(\text{cos} \theta\), is one of the fundamental trigonometric functions. It relates the angle \(\theta\) to the x-coordinate of a point on the unit circle. Essentially, for any given angle, the cosine function outputs the x-component of the corresponding point on the circle.
  • At \(\theta = 0\), \(\text{cos} \theta\) is 1 because the point \((1, 0)\) lies on the x-axis.
  • At \(\theta = \frac{\pi}{2}\), or 90 degrees, \(\text{cos} \theta\) is 0 because the point \((0, 1)\) lies on the y-axis where there's no x-component.
  • The cosine value ranges between -1 and 1 as \(\theta\) varies.
Understanding where \(\text{cos} \theta\) equals zero helps to solve equations involving the cosine function. The zero values occur where the angle represents a point purely on the y-axis.
unit circle
The unit circle is a crucial concept in trigonometry. It's a circle with a radius of 1, centered at the origin of a coordinate plane. Each point on the unit circle can be described by coordinates \((\text{cos} \theta, \text{sin} \theta)\), where \(\theta\) is the angle formed with the positive x-axis.
  • This circle helps visualize the values of trigonometric functions such as sine, cosine, and tangent.
  • For any angle \(\theta\), the x-coordinate corresponds to \(\text{cos} \theta\), and the y-coordinate corresponds to \(\text{sin} \theta\).
  • Key points where \(\text{cos} \theta = 0\) are \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\), which lie on the y-axis.
When solving for \(\text{cos} \theta = 0\), these points on the unit circle help identify the specific angles and their general solutions.
general solutions
General solutions in trigonometry provide formulas that encompass all possible solutions to a trigonometric equation. For instance, to solve \(\text{cos} \theta = 0\), you need to find where the cosine function equals zero.
  • Using the unit circle, you identify that cosine is zero at angles \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\).
  • To generalize this, note that cosine will repeat every \(\pi\) units (half-circles).
  • Thus, the general formula is \(\theta = \frac{\pi}{2} + k\pi\), where \(\theta\) can be any angle that fits this pattern and \(\text{k}\) is an integer.
Applying this formula allows you to list specific angles that satisfy \(\text{cos} \theta = 0\), such as \(\frac{\pi}{2}\), \(\frac{3\pi}{2}\), and others by simply plugging in different integer values of \(\text{k}\). This helps in finding any number of solutions, not just one specific angle.

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