Question

9.

a. \\(\sin\theta = \\)

b. \\(w = \\)

c. \\(\theta =

1. describe the angles of rotation that make the \\(y\\)-values of the points be positive and the angles of rotation that make the \\(y\\)-values be negative.

1. what do you notice about the \\(y\\)-values and the value of sine in the previous graphs?

1. in the graph shown, the radius of the circle is 1 unit. the intersections of the circle and the axes are labeled. based on your observation in #11, what do you think the value of sine might be for the following values of \\(\theta\\)?

a. \\(90^\circ\\)

b. \\(180^\circ\\)

c. \\(270^\circ\\)

Explanation:

Question 10

To determine when \( y \)-values are positive or negative in a unit circle (or any circle with angle rotation), we use the quadrant system. Angles are measured counterclockwise from the positive \( x \)-axis.

• Positive \( y \)-values: In the unit circle, the \( y \)-coordinate of a point \((x, y)\) on the circle (corresponding to angle \( \theta \)) is positive when the point is in Quadrants I and II. This occurs for angles \( \theta \) where \( 0^\circ < \theta < 180^\circ \) (or \( 0 < \theta < \pi \) radians).
• Negative \( y \)-values: The \( y \)-coordinate is negative when the point is in Quadrants III and IV. This occurs for angles \( \theta \) where \( 180^\circ < \theta < 360^\circ \) (or \( \pi < \theta < 2\pi \) radians). Additionally, angles coterminal with these (adding/subtracting \( 360^\circ \) or \( 2\pi \)) follow the same pattern.

In the context of the unit circle (or any circle used to model trigonometric functions), the \( y \)-value of a point on the circle corresponding to an angle \( \theta \) (measured from the positive \( x \)-axis) is equal to the sine of that angle. Mathematically, for a point \((x, y)\) on the unit circle (radius \( r = 1 \)), \( y = r\sin\theta \). When \( r = 1 \) (unit circle), \( y = \sin\theta \). For non - unit circles, the \( y \)-coordinate of a point at angle \( \theta \) with radius \( r \) is \( y = r\sin\theta \), so the \( y \)-value is proportional to \( \sin\theta \) (with the proportionality constant being the radius \( r \)).

Step 1: Identify the point on the unit circle.

For \( \theta = 90^\circ \) (or \( \frac{\pi}{2} \) radians), the point on the unit circle is \( (0, 1) \).

Step 2: Relate \( y \)-value to \( \sin\theta \).

From Question 11, \( \sin\theta=y \) (for unit circle, \( r = 1 \)). The \( y \)-value of the point \( (0, 1) \) is \( 1 \), so \( \sin(90^\circ)=1 \).

Answer:

• Angles with positive \( y \)-values: Angles of rotation \( \boldsymbol{\theta} \) where \( 0^\circ < \theta < 180^\circ \) (or \( 0 < \theta < \pi \) radians, and all coterminal angles by adding/subtracting \( 360^\circ n \), \( n \in \mathbb{Z} \)).
• Angles with negative \( y \)-values: Angles of rotation \( \boldsymbol{\theta} \) where \( 180^\circ < \theta < 360^\circ \) (or \( \pi < \theta < 2\pi \) radians, and all coterminal angles by adding/subtracting \( 360^\circ n \), \( n \in \mathbb{Z} \)).

Question 11