Problem 27 - Contemporary Precalculus

Question

Find all angles \(\theta\) with \(0^{\circ} \leq \theta<360^{\circ}\) that are solutions of the given equation. IHint: Put your calculator in degree mode and replace \(\pi\) by \(180^{\circ}\) in the solution algorithms for basic equations. \(]\) $$\cos \theta=-.42$$

Step-by-Step Solution

Answer

Question: Find all angles between \(0^{\circ}\) and \(360^{\circ}\) that have a cosine of -0.42. Answer: The angles that satisfy this condition are \(114.62^{\circ}\) and \(245.38^{\circ}\).

Step 1: Determine the reference angle (acute angle)

To find the first angle with a cosine of -0.42, we'll use the inverse cosine function and find the reference angle (acute angle). Since the cosine is negative, we will take the absolute value of -0.42 and find the reference angle (acute angle) with that: $$\cos^{-1}(0.42)=65.38^{\circ}$$ This gives us the reference angle, which we'll use to find the angles in the second and third quadrants.

Step 2: Find the angle in the second quadrant

Since cosine is negative in the second quadrant and we know the reference angle, we can find the angle in the second quadrant by subtracting the reference angle from 180: $$\theta_{2} = 180^{\circ} - 65.38^{\circ} = 114.62^{\circ}$$ So, one of the angles with a cosine of -0.42 is \(114.62^{\circ}\).

Step 3: Find the angle in the third quadrant

Similarly, to find the angle in the third quadrant, we'll add the reference angle to 180: $$\theta_{3} = 180^{\circ} + 65.38^{\circ} = 245.38^{\circ}$$ Hence, the second angle with a cosine of -0.42 is \(245.38^{\circ}\).

Step 4: Confirm the angles

After finding the two angles of \(\theta\) (\(114.62^{\circ}\) and \(245.38^{\circ}\)) with a cosine of -0.42, we double-check the results using a calculator to confirm they meet the requirements. $$\cos(114.62^{\circ}) \approx -0.42$$ $$\cos(245.38^{\circ}) \approx -0.42$$ Since the values match, the angles we found are the correct solutions. The angles \(\theta\) with \(0^{\circ} \leq \theta<360^{\circ}\) that satisfy the equation \(\cos \theta=-0.42\) are \(114.62^{\circ}\) and \(245.38^{\circ}\).

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that relates the angle of a right-angled triangle to the lengths of the sides. It is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. For an angle \( \theta \), this ratio is written as \( \cos(\theta) \).

In the context of the unit circle, \( \cos(\theta) \), represents the x-coordinate of the point where the terminal side of the angle intersects the circle. The cosine function has a range of [-1, 1], which means it oscillates between these values as the angle \( \theta \) increases from \(0^\circ \) to \(360^\circ\).

When solving trigonometric equations such as \( \cos \theta = -.42 \), the cosine function helps identify the angle \( \theta \) whose cosine is -0.42. Since the cosine function is even—meaning it has symmetry about the y-axis—the same absolute value of cosine corresponds to two angles in different quadrants, one on each side of the y-axis, within a single cycle of 360 degrees.

Reference Angle

The concept of a reference angle is essential in trigonometry to simplify the process of finding the angles that satisfy trigonometric equations. A reference angle is the smallest acute angle (an angle less than \(90^\circ\)) that is formed by the terminal side of an angle and the horizontal axis.

It's crucial in the context of the unit circle. Regardless of the actual angle's quadrant, the reference angle shares the same sine, cosine, and tangent values, except possibly for a difference in sign. When working with negative cosine values, as the cosine function is negative in the second and third quadrants, the reference angle provides a means to systematically find the possible solutions.

By first using the inverse cosine function on the absolute value of the cosine, as in the step-by-step solution \( \cos^{-1}(0.42) \approx 65.38^\circ \), we find the reference angle which is then used to determine the respective angles in the appropriate quadrants where the cosine value is negative.

Inverse Trigonometric Functions

Inverse trigonometric functions, also known as arc functions, allow us to find an angle when given the value of a trigonometric function. For instance, \( \cos^{-1} \) is the inverse function of the cosine function, and its purpose is to determine the angle whose cosine is a given number.

However, because of the periodic nature of trigonometric functions, a particular trigonometric value corresponds to multiple angles. The inverse trigonometric functions, by convention, return the principal value — this value lies within the range of angles where each function is defined to be injective. For the cosine function, the principal value returned by \( \cos^{-1} \) is between \(0^\circ\) and \(180^\circ\), which will always yield an acute reference angle.

To apply inverse trigonometric functions like \( \cos^{-1} \) in solving the given equation \( \cos \theta = -.42 \), we first determine the reference angle. Then we find the corresponding angles in the specific quadrants where the cosine is negative. This approach is illustrated in the step-by-step solution, showing the versatility and importance of inverse trigonometric functions in solving trigonometric equations.