Question

In Exercises \(31-36,\) solve the equation in the specified interval. $$\cos x=-0.7\( \)2 \pi \leq x<4 \pi$$

Short answer

The solutions to the equation in the specified interval are approximately \(5.4880\) and \(7.0783\) radians.

Step-by-step solution

Identify the Quadrants

Since \(\cos x=-0.7\), x is in either the second or third quadrants because cosine is negative in those quadrants (according to the unit circle).

Find Related Acute Angle

Find the associated acute angle (a) by setting cosine positive and using the inverse cosine function. Solve \(a=\cos^{-1}(0.7)\) to get \(a \approx 0.7954\) radian.

Get Solutions for Second and Third Quadrants

Since cosine is negative in the second and third quadrants, find the solutions for these. For the second quadrant, use the formula \( \pi - a\) and for the third quadrant use the formula \( \pi + a\). This gives \( x_2 = \pi - a \approx 2.3462\) and \( x_3 = \pi + a \approx 3.9370\).

Check If Solutions are in the Given Interval

Check if these solutions fall within the given range \(2 \pi \leq x<4 \pi\). \( x_2\) is indeed less than \(2 \pi\), and \(x_3\) is less than \(4 \pi\), hence, they don't fall within the assigned interval.

Solve for Other Solutions in the Interval

Now, there should exist other solutions in the interval. Since cosine function has a period of \(2 \pi\), there are two solutions in every period. The solutions in the given interval can be find by adding multiples of \(2 \pi\) to the found solutions. Add \(2 \pi\) to both \(x_2\) and \(x_3\) to get solutions in the given range. This gives \( x_2' = x_2 + 2 \pi \approx 5.4880\) and \( x_3' = x_3 + 2 \pi \approx 7.0783\).

Conclusion

Therefore, the solutions to the equation \(\cos x=-0.7\) in the interval \(2 \pi \leq x<4 \pi\) are \(x_2'\) and \(x_3'\) which are approximately \(5.4880\) and \(7.0783\) radians.

Key concepts

Unit Circle

The unit circle is a foundational concept in trigonometry, acting as a visual tool to understand angles and trigonometric functions. Imagine a circle with a radius of one unit centered at the origin of a coordinate plane. By drawing a line from the center to any point on the circumference, we create an angle with the x-axis. The x-coordinate of this point gives us the cosine of the angle, and the y-coordinate provides the sine of the angle.

When we're solving trigonometric equations, such as \( \cos x=-0.7 \), we rely on the unit circle to determine in which quadrants the angle \( x \) lies. Since cosine corresponds to the x-coordinate, a negative cosine value means that the terminal side of the angle \( x \) is in either quadrant II or III where the x-coordinates are negative. This information is crucial for finding the correct solutions to trigonometric equations within a specified interval.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to work backward from the trigonometric values to find the angles they represent. They are the inverses of the sine, cosine, and tangent functions and are denoted as \( \sin^{-1} \), \( \cos^{-1} \), and \( \tan^{-1} \), respectively. When the equation \( \cos x=-0.7 \) appears, we can use \( \cos^{-1} \) to find the related acute angle \( a \) that has a positive cosine value.

To continue with the problem-solving, we calculate \( a=\cos^{-1}(0.7) \) to find the angle in radians that corresponds to the cosine of 0.7. Remember that \( \cos^{-1} \) will always give us the smallest angle whose cosine matches the given value, which is why it's considered an acute angle. This step sets the stage for determining the solution to the trigonometric equation in specific quadrants.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all allowed values of the variables involved. They are used to simplify expressions, solve trigonometric equations, and transform one form of trigonometric expression into another. Common identities include Pythagorean identities, angle sum and difference identities, and double angle formulas, among others.

In our example, while we haven't used a specific trigonometric identity, the concept is still relevant. Knowledge of identities can provide alternative methods for approaching trigonometric equations and determining angles related to the specific functional values. Understanding identities is central to mastery of trigonometry, aiding in simplification, conversion of trigonometric forms, and solutions for various domains and intervals.

Periodicity of Trigonometric Functions

Trigonometric functions are periodic, meaning they repeat their values in regular intervals along the domain. The period is the length of the smallest interval over which the function repeats. For the cosine function, the period is \( 2\pi \) radians. This periodic nature is pivotal when solving trigonometric equations over a specified interval, especially when the initial solutions do not fall within the desired range.

In the equation \( \cos x=-0.7 \) with an interval from \( 2\pi \) to \( 4\pi \) radians, the periodicity concept guides us to find additional solutions. By adding multiples of the period \( 2\pi \) to initial solutions, we can locate the corresponding angles that lie within the given interval. Thus, understanding the periodicity of trigonometric functions enables us to fully solve equations over any range.