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Consider the angle shown above measured (in radians) counterclockwise from an initial ray pointing in the 3-o'clock direction to a terminal ay pointing from the origin to (1.26, -2.39). What is the measure of 0 (in radians)? = Y 0 x Ⓡ (1.26,-2.39) Preview

Consider the angle shown above measured (in radians) counterclockwise from an initial ray pointing in the 3-o'clock direction to a terminal ay pointing from the origin to (1.26, -2.39). What is the measure of 0 (in radians)? = Y 0 x Ⓡ (1.26,-2.39) Preview

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### Educational Content: Understanding Radian Measure in Trigonometry #### Diagram Analysis: The diagram illustrates a circle on a coordinate plane, centered at the origin (0,0). It features: - **Axes**: Horizontal (x-axis) and vertical (y-axis) are shown, intersecting at the origin. - **Circle**: A blue circle centered at the origin with its circumference passing through crucial directional points. - **Angle θ**: An angle is indicated in the circle by a gray arc, starting from the positive x-axis (3-o'clock direction) and rotating counterclockwise to a terminal ray. - **Terminal Ray**: A black arrow extending from the origin to the point (1.26, -2.39) in the fourth quadrant. - **Coordinates**: The point (1.26, -2.39) is marked in purple, indicating the terminal point of the angle's ray. #### Problem Statement: Consider the angle \( \theta \) shown above, measured in radians, counterclockwise from an initial ray pointing in the 3-o’clock direction to a terminal ray pointing from the origin to (1.26, -2.39). What is the measure of \( \theta \) (in radians)? \[ \theta = \quad \text{[input box]} \quad \text{Preview button} \] #### Explanation: In trigonometry, angles are often measured in radians, where the complete circle of 360 degrees is equivalent to \( 2\pi \) radians. Here, the task is to determine the radian measure of the angle \( \theta \) based on the coordinates given for the terminal point. The angle is formed in standard position with its vertex at the origin, its initial side along the positive x-axis, and sweeps out to its terminal side at the specified coordinates. To solve this, use trigonometric relationships from the coordinates provided to calculate \( \theta \), involving the arctangent or other relevant trigonometric functions.
Transcribed Image Text:### Educational Content: Understanding Radian Measure in Trigonometry #### Diagram Analysis: The diagram illustrates a circle on a coordinate plane, centered at the origin (0,0). It features: - **Axes**: Horizontal (x-axis) and vertical (y-axis) are shown, intersecting at the origin. - **Circle**: A blue circle centered at the origin with its circumference passing through crucial directional points. - **Angle θ**: An angle is indicated in the circle by a gray arc, starting from the positive x-axis (3-o'clock direction) and rotating counterclockwise to a terminal ray. - **Terminal Ray**: A black arrow extending from the origin to the point (1.26, -2.39) in the fourth quadrant. - **Coordinates**: The point (1.26, -2.39) is marked in purple, indicating the terminal point of the angle's ray. #### Problem Statement: Consider the angle \( \theta \) shown above, measured in radians, counterclockwise from an initial ray pointing in the 3-o’clock direction to a terminal ray pointing from the origin to (1.26, -2.39). What is the measure of \( \theta \) (in radians)? \[ \theta = \quad \text{[input box]} \quad \text{Preview button} \] #### Explanation: In trigonometry, angles are often measured in radians, where the complete circle of 360 degrees is equivalent to \( 2\pi \) radians. Here, the task is to determine the radian measure of the angle \( \theta \) based on the coordinates given for the terminal point. The angle is formed in standard position with its vertex at the origin, its initial side along the positive x-axis, and sweeps out to its terminal side at the specified coordinates. To solve this, use trigonometric relationships from the coordinates provided to calculate \( \theta \), involving the arctangent or other relevant trigonometric functions.
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