Problem 111 - A Graphical Approach to Precalculus with Limits

Question

Find all values of \(\theta\) if \(\theta\) is in the interval \(\left[0^{\circ}, 360^{\circ}\right)\) and has the given function value. Give calculator approximations to as many decimal places as your calculator displays. $$\cos \theta=0.68716510$$

Step-by-Step Solution

Answer

\( \theta \approx 46.498^{\circ} \) and \( \theta \approx 313.502^{\circ} \).

Step 1: Use Inverse Cosine Function

To find the initial angle that corresponds to the given cosine value, use the inverse cosine function. Calculate \( \theta = \cos^{-1}(0.68716510) \). Use a calculator to find \( \theta \).

Step 2: Find Secondary Angle

Since the cosine function is positive in the first and fourth quadrants, the other angle that has the same cosine value can be found by using \( \theta = 360^{\circ} - \theta_1 \), where \( \theta_1 \) is the angle found in Step 1.

Step 3: Verify the Range

Ensure that all angle solutions are within the given range \([0^{\circ}, 360^{\circ})\). Both angles should already lie within this range by construction.

Step 4: Calculate using Calculator

Using a calculator, compute the value for \( \theta \). You will obtain \( \theta_1 \approx 46.498^{\circ} \). The secondary angle is then \( \theta_2 = 360^{\circ} - 46.498^{\circ} \approx 313.502^{\circ} \). Ensure these values closely match your calculator's output.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles based on given trigonometric values. In this particular exercise, we deal with the inverse cosine function, which is noted as \(\cos^{-1}(x)\). This function helps us find an angle whose cosine is given.
In general, the ranges of the inverse cosine function are restricted. For angles, \(\cos^{-1}(x)\) outputs values between \(0^{\circ}\) and \(180^{\circ}\) when dealing with degrees. This means if we take \(\cos^{-1}(0.68716510)\), we'll initially find an angle located in the first quadrant.
Using this function, we can determine our first angle that has a cosine of \(0.68716510\), which is the starting point to solving trigonometric equations. Remember always to use a calculator for precise results.

Cosine Function

The cosine function is one of the primary trigonometric functions, alongside sine and tangent. It relates the angle in a right triangle to the ratio of the adjacent side over the hypotenuse.
In the context of the unit circle, which has a radius of 1, the cosine of an angle \(\theta\) corresponds to the x-coordinate of the point where the terminal side of the angle intersects the unit circle.

• For example, if \(\cos(\theta) = 0.68716510\), it means the x-coordinate of our point is approximately \(0.68716510\).

As a periodic function, cosine repeats its values every \(360^{\circ}\). This cyclic nature helps us find multiple angles that have the same cosine value within a 360-degree interval.

Angle Calculation

Calculating angles from a cosine value involves a few key steps. First, we use the inverse cosine function to find the "primary angle." This is the basic angle in the range of \([0^{\circ}, 180^{\circ}]\).
In the exercise problem, we calculate \(\theta_1 = \cos^{-1}(0.68716510)\), which gives approximately \(46.498^{\circ}\). This angle lies in the first quadrant as expected, due to the positive cosine value.
By understanding the periodic nature of cosine, we also learn that another angle with the same cosine exists in the fourth quadrant. To find this "secondary angle," we use the equation \(\theta_2 = 360^{\circ} - \theta_1\), leading us to approximately \(313.502^{\circ}\). Calculators are often needed for precision when evaluating these angles.

Quadrants

The concept of quadrants is essential to solving trigonometric equations. The coordinate plane is divided into four quadrants, each representing potential angle locations:

• Quadrant I: Angles from \(0^{\circ}\) to \(90^{\circ}\)
• Quadrant II: Angles from \(90^{\circ}\) to \(180^{\circ}\)
• Quadrant III: Angles from \(180^{\circ}\) to \(270^{\circ}\)
• Quadrant IV: Angles from \(270^{\circ}\) to \(360^{\circ}\)

In this problem, since \(\cos(\theta)\) is positive, \(\theta\) is located in Quadrants I and IV.
The first solution, \(\theta_1 \approx 46.498^{\circ}\), lies in Quadrant I. The second solution, calculated as \(\theta_2 \approx 313.502^{\circ}\), resides in Quadrant IV. Knowing the properties of the quadrants helps us predict where angles will land and solve equations accurately.