Problem 29 - Algebra 2 and Trigonometry

Question

In \(28-43,\) for each function value, if \(0^{\circ} \leq \theta <3 60^{\circ},\) find, to the nearest degree, two values of \(\theta\) \(\cos \theta=0.6283\)

Step-by-Step Solution

Answer

Two values for \( \theta \) are \( 51^{\circ} \) and \( 309^{\circ} \).

Step 1: Understanding the Problem

We need to find two values of \( \theta \) where \( \cos \theta = 0.6283 \) and \( 0^{\circ} \leq \theta < 360^{\circ} \). To solve this, we need to use the cosine inverse function and knowledge of the cosine graph within the specified interval.

Step 1: Finding the First Value using Inverse Cosine

Use the inverse cosine function to find one value of \( \theta \). This is done by calculating \( \theta_1 = \cos^{-1}(0.6283) \). Use a calculator to find an approximate value: \( \theta_1 \approx 51^{\circ} \).

Step 2: Finding the Second Value using Cosine Symmetry

The cosine function is symmetric around 0, and it repeats every \( 360^{\circ} \). Since cosine is positive in the first and fourth quadrants, the second value \( \theta_2 \) can be found using \( \theta_2 = 360^{\circ} - \theta_1 \). Thus, \( \theta_2 = 360^{\circ} - 51^{\circ} = 309^{\circ} \).

Step 3: Confirming Values are Within Range

Both values, \( \theta_1 = 51^{\circ} \) and \( \theta_2 = 309^{\circ} \), are within the specified range \( 0^{\circ} \leq \theta < 360^{\circ} \), confirming they are valid solutions.

Key Concepts

Inverse Cosine

The inverse cosine, often denoted as \( \cos^{-1} \) or arccos, is a function used to find angles when the value of the cosine function is known. In simple terms, if you have the cosine of an angle and you want to find the angle itself, you need to use the inverse cosine. For example, in our problem, we have the value \( \cos \theta = 0.6283 \).

To find the angle \( \theta \), we use \( \theta = \cos^{-1}(0.6283) \). This calculation tells us the angle at which the cosine has this specific value. Remember,

• the result of the inverse cosine is always in the range \( 0^{\circ} \) to \( 180^{\circ} \),
• for angles specifically in degrees, a calculator is usually required.

The inverse cosine helps us find angles when dealing with trigonometric equations, making it a powerful tool in our mathematical toolkit.

Cosine Symmetry

The symmetry of the cosine function is an interesting property that often helps simplify trigonometric calculations. Cosine symmetry refers to how the cosine function behaves with respect to rotations and reflections on the unit circle.

Understanding this symmetry is crucial for identifying the two values of \( \theta \) that have the same cosine value. Cosine values repeat every \( 360^{\circ} \) since they are periodic with this interval. Within one full rotation of \( 0^{\circ} \) to \( 360^{\circ} \), cosine is:

• positive in the first quadrant \( 0^{\circ} \text{ to } 90^{\circ} \)
• also positive again in the fourth quadrant \( 270^{\circ} \text{ to } 360^{\circ} \)

To find the second angle \( \theta_2 \), corresponding to the same cosine value, we use the property: \( \theta_2 = 360^{\circ} - \theta_1 \). This second angle exists due to the symmetry and repetition of the cosine function on the unit circle.

Trigonometric Graph Analysis

Analyzing trigonometric graphs provides deep insight into how trigonometric functions behave and change over their domain. The cosine graph, in particular, is a wave-like curve that oscillates between \(-1\) and \(1\), repeating every \(360^{\circ}\) or \(2\pi\) radians, which is known as the period of the cosine function.

When examining these graphs, we can determine values of angles for given cosine outputs, hence solving trigonometric equations effectively. Key aspects of the cosine graph include:

• Amplitude: the peak value of \(1\) when at its maximum, and \(-1\) when at its minimum.
• Period: every \(360^{\circ} \), the graph repeats.
• Axis symmetry: the cosine wave is symmetric around the vertical axis, making it easier to find multiple angles with the same cosine.

Utilizing graph analysis can vastly improve problem-solving skills in trigonometry by providing visual understanding alongside algebraic manipulation.