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Chapter 6: Problem 69

Find the exact values of 6 in the given interval that satisfy the given condition. $$[0,2 \pi) ; \quad \cos ^{2} s=\frac{1}{2}$$

Short Answer

Expert verified
The exact values of \( s \) are \( \frac{ \pi }{4 }, \frac{ 3 \pi }{4 }, \frac{ 5 \pi }{4 } \) and \( \frac{ 7 \pi }{4 } \).

Step by step solution

01

Simplify the given equation

Given the equation \( \cos^2(s) = \frac{1}{2} \), we need to simplify it to find the value of \( \cos(s) \). Taking the square root of both sides, we get \( \cos(s) = \pm \frac{1}{ \sqrt{2} } \) which simplifies to \( \cos(s) = \pm \frac{ \sqrt{2} }{2 } \).
02

Identify the reference angles

The values of \( \cos(s) = \pm \frac{ \sqrt{2} }{2 } \) correspond to the angles \( s = \frac{ \pi }{4 } \) and \( s = - \frac{ \pi }{4 } \).
03

Find all solutions in the interval

Since \( \cos(s) \) is periodic with period \( 2 \pi \), the angles within the interval \( [0, 2 \pi) \) can be found by adding multiples of \( 2 \pi \) to the reference angles: \ \ For \( \cos(s) = \frac{ \sqrt{2} }{2 } \), the angles are \( \frac{ \pi }{4 } \) and \( \frac{ 7 \pi }{4 } \) \ \ For \( \cos(s) = - \frac{ \sqrt{2} }{2 } \), the angles are \( \frac{ 3 \pi }{4 } \) and \( \frac{ 5 \pi }{4 } \)

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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, usually denoted as \( \text{cos}(x) \, \), is one of the primary trigonometric functions. It measures the horizontal distance from the origin to the point on the unit circle corresponding to a given angle. The cosine function fluctuates between -1 and 1 as the angle increases from 0 to \( 2\text{蟺} \) radians.
It's useful to remember that:
  • \( \text{cos}(0) = 1 \, \)
  • \( \text{cos}(\text{蟺}/2) = 0 \, \)
  • \( \text{cos}(\text{蟺}) = -1 \)
These values provide reference points when solving trigonometric equations.
Reference Angles
Reference angles are key to solving many trigonometric equations. A reference angle is the acute angle that an angle in standard position makes with the x-axis. For example, for the angles where \( \text{cos}(s) = \frac{ \sqrt{2} }{2 } \, \), the primary reference angles are \( \frac{ \text{蟺} }{4} \, \) and \( - \frac{ \text{蟺} }{4} \, \).
With the cosine function, these angles imply:
  • \( s \, \) lies in the first quadrant for \( + \frac{ \text{蟺} }{4} \, \)
  • \( s \, \) in the fourth quadrant for \( - \frac{ \text{蟺} }{4} \)
Understanding reference angles helps you find all possible solutions for \( s \) within a given interval.
Periodicity of Trigonometric Functions
The cosine function exhibits periodicity, meaning it repeats its values at regular intervals. The period of the cosine function is \( 2\text{蟺} \, \). This characteristic lets you extend solutions to trigonometric equations by adding multiples of \( 2\text{蟺} \).
For the equation \( \text{cos}(s) = \frac{ \sqrt{2} }{2 } \, \), knowing the basic angles \( \text{蟺}/4 \, \), \( 3\text{蟺}/4 \, \), \( 5\text{蟺}/4 \, \), and \( 7\text{蟺}/4 \) helps you find all solutions in the interval \( [0, 2\text{蟺}) \).
The periodicity lets us handle cases like:
  • Adding \( 2\text{蟺} \, \) to a reference angle
  • Identifying equivalent angles across revolution cycles
Interval Notation
When working with trigonometric equations, it's common to express solutions within specific intervals using interval notation. For the interval \( [0, 2\text{蟺}) \),
  • The square bracket \( [ \) at 0 means 0 is included in the interval.
  • The parenthesis \( ) \) at \( 2\text{蟺} \) indicates \( 2\text{蟺} \) itself is not included.
Interval notation provides a clear and concise way to denote the set of solutions for \( s \) within given bounds.
In our example, the solutions \( \frac{ \text{蟺} }{4 } \, \), \( \frac{ 3\text{蟺} }{4} \, \), \( \frac{ 5\text{蟺} }{4} \, \), and \( \frac{ 7\text{蟺} }{4} \) are within the interval \( [0, 2\text{蟺}) \), ensuring all possible values are represented accurately.

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