Textbook Question
Find each exact function value. See Example 3.
sin (-8π/ 3)
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1. Measuring Angles
Complementary and Supplementary Angles
Problem 90
Textbook Question
Give an expression that generates all angles coterminal with an angle of π/2 radians. Let n represent any integer.
Verified step by step guidance1
Recall that angles are coterminal if they differ by an integer multiple of a full rotation. In radians, a full rotation is \(2\pi\).
Given the angle \(\frac{\pi}{2}\), to find all angles coterminal with it, we add integer multiples of \(2\pi\) to this angle.
Express this mathematically as \(\theta = \frac{\pi}{2} + 2\pi n\), where \(n\) is any integer (positive, negative, or zero).
This expression generates all angles that share the same terminal side as \(\frac{\pi}{2}\) when drawn in standard position.
Thus, the general formula for all coterminal angles with \(\frac{\pi}{2}\) is \(\theta = \frac{\pi}{2} + 2\pi n\), with \(n \in \mathbb{Z}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Coterminal Angles
Coterminal angles are angles that share the same initial and terminal sides but differ by full rotations. In radians, adding or subtracting multiples of 2π to an angle results in coterminal angles. This concept helps identify all angles equivalent in position on the unit circle.
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Coterminal Angles
Angle Measurement in Radians
Radians measure angles based on the radius of a circle, where 2π radians equal one full rotation (360 degrees). Understanding radians is essential for expressing angles and their coterminal counterparts in a mathematically consistent way.
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General Expression for Coterminal Angles
The general formula for coterminal angles is θ + 2πn, where θ is the initial angle and n is any integer. This expression generates all angles coterminal with θ by adding full rotations, allowing for a complete set of equivalent angles.
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