Problem 75 - Precalculus

Question

Find all possible values of \(\theta,\) where \(0^{\circ} \leq \theta \leq 360^{\circ}\) $$\sin \theta=-1$$

Step-by-Step Solution

Answer

The angle \( \theta \) is \( 270^{\circ} \).

Step 1: Understanding the Sine Function

Recall that the sine function, \( \sin \theta \), gives the y-coordinate of a point on the unit circle corresponding to the angle \( \theta \). The range of the sine function is between -1 and 1 inclusive.

Step 2: Identifying the Condition

We need to find all angles \( \theta \) such that \( \sin \theta = -1 \). This condition corresponds to the point where the y-coordinate is at its minimum possible value on the unit circle.

Step 3: Finding the Angle

The angle \( \theta \), where \( \sin \theta = -1 \), occurs at the bottom of the unit circle. This is at \( \theta = 270^{\circ} \).

Step 4: Conclusion

Given that \( 0^{\circ} \leq \theta \leq 360^{\circ} \), we conclude that the only angle in this range for which \( \sin \theta = -1 \) is \( 270^{\circ} \).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one unit, centered at the origin of a coordinate plane (0,0). This circle is particularly useful for defining the trigonometric functions for all angles.
Within this circle, each point (x,y) corresponds to an angle \( \theta \) measured from the positive x-axis. The coordinates themselves represent the cosine and sine values of the angle \( \theta \), where \( x = \cos \theta \) and \( y = \sin \theta \).

• The unit circle allows us to visualize angles and their trigonometric values.
• Every angle \( \theta \) can be represented on the unit circle, within the interval from \( 0^{\circ} \) to \( 360^{\circ} \), or equivalently between \( 0 \) and \( 2\pi \) radians.
• This makes it much easier to find exact values of sine and cosine for standard angles.

Sine Function

The sine function, often notated as \( \sin(\theta) \), provides the y-coordinate of a point on the unit circle corresponding to an angle \( \theta \). Knowing this helps in determining how high or low the point is relative to the center line (x-axis).

• The sine function oscillates between -1 and 1. Its graph is a smooth wave-like pattern, known as a sine wave.
• The maximum value of \( \sin \theta \) is 1, which occurs when the angle is \( 90^{\circ} \) (or \( \frac{\pi}{2} \) radians), placing the point directly at the top of the unit circle.
• The minimum value of \( \sin \theta \) is -1, happening at \( 270^{\circ} \) (or \( \frac{3\pi}{2} \) radians), when the point is directly at the bottom of the unit circle. This vertical symmetry helps us easily find when sine takes on its extreme values, as in the problem "\( \sin \theta = -1 \)."

Angle Measurement

Angles can be measured in either degrees or radians, with degrees being the more commonly used measure in everyday applications. In mathematical contexts, both systems are crucial.

• Degrees divide a full circle into 360 equal parts. This is quite intuitive and widely understood across various fields.
• Radians divide the circle differently, with one radian being the angle made by takin all radius in circle arc equal to the radius length. There are \( 2\pi \) radians in a circle, equating to 360^{\circ}
• Since \( \sin(\theta) = -1 \) only occurs at \( 270^{\circ} \), understanding the connection between radians and degrees helps us easily convert or verify the angle in both systems.