Problem 74 - Precalculus

Question

Find all possible values of \(\theta,\) where \(0^{\circ} \leq \theta \leq 360^{\circ}\) $$\sin \theta=0$$

Step-by-Step Solution

Answer

The values of \( \theta \) are \( 0^{\circ}, 180^{\circ}, \text{and} 360^{\circ} \).

Step 1: Understand the solution for sine

The sine of an angle 94equals zero when the angle itself is a multiple of 9018090degrees. This is because the sine function corresponds to the y-coordinate of a point on the unit circle, and points where this y-coordinate is zero are at the angles 090degrees, 18090degrees, and 36090degrees.

Step 2: Convert the multiples of 180

Given that sine is zero at multiples of 9018090degrees, we express the multiples within the specified range: 90090degrees, 18090degrees, and 36090degrees.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. The significance of the unit circle is that it provides a simple way to understand and visualize trigonometric functions.

Here's why the unit circle is important:

• The x-coordinate of any point on the unit circle represents the cosine of the angle formed with the positive x-axis.
• The y-coordinate of the same point tells us the sine of the angle.
• Angles in the unit circle are typically measured from the positive x-axis, and they wrap counter-clockwise around the circle.

Knowing this, when exploring the sine function, you are essentially tracking the y-coordinate as you move around the circle. So, every time the y-coordinate is zero, the sine of that angle is also zero.

Sine Function

The sine function is a periodic function that is crucial in trigonometry and is closely tied to the unit circle. Let's break it down:

• The sine function, \sin(\theta), gives the y-coordinate of a point on the unit circle at an angle \( \theta \) from the positive x-axis.
• Its values oscillate between -1 and 1, reflecting the highest and lowest points on the unit circle.
• Sine is periodic, with a key period of \( 360^{\circ} \) in degrees. This means it repeats its values for every full circle around the unit circle.

For the problem of finding when \( \sin(\theta) = 0 \), remember:

• This occurs when the y-coordinate is zero, which happens when \( \theta \) is at \( 0^{\circ}, 180^{\circ}, \) and \( 360^{\circ} \).

These angles represent the points where the circle intersects the x-axis.

Angles in Degrees

Angles are measures of rotation that can be expressed in different units: degrees and radians. In many trigonometry problems, including the one here, degrees are used. Here's why understanding angles in degrees is helpful:

• Degrees measure angles in a circle, with a full circle being \( 360^{\circ} \).
• Each quadrant in a unit circle, therefore, spans \( 90^{\circ} \), making it easier to divide and analyze different angles.
• Common angles like \( 0^{\circ}, 90^{\circ}, 180^{\circ}, \) and \( 270^{\circ} \) provide intuitive landmarks on the unit circle for sine and cosine values.

In solving trigonometric equations, expressing angles in degrees can make it easier to understand and visualize geometric relations and periodic functions, especially when dealing with circles and rotational problems.