Textbook Question
Find a calculator approximation to four decimal places for each circular function value. See Example 3. csc (―9.4946)
3. Unit Circle
Trigonometric Functions on the Unit Circle
Problem 56
Textbook Question
Without using a calculator, decide whether each function value is positive or negative. (Hint: Consider the radian measures of the quadrantal angles, and remember that π ≈ 3.14.)
sin ( ―1)
Verified step by step guidance1
Identify the angle given: the problem asks about \( \sin(-1) \), where \(-1\) is in radians.
Recall that \( -1 \) radian is a negative angle, which means it is measured clockwise from the positive x-axis.
Determine the quadrant where the angle \( -1 \) radian lies. Since \( \pi \approx 3.14 \), and \( -1 \) is between \( 0 \) and \( -\pi/2 \) (which is approximately \(-1.57\)), the angle lies in the fourth quadrant.
Recall the sign of sine in each quadrant: sine is positive in the first and second quadrants, and negative in the third and fourth quadrants.
Since \( -1 \) radian is in the fourth quadrant, \( \sin(-1) \) is negative.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Understanding Radian Measure
Radian measure is a way to express angles based on the radius of a circle. One full circle is 2π radians, so π radians equals 180 degrees. Knowing the approximate value of π (about 3.14) helps to locate angles on the unit circle and determine their quadrant.
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Quadrantal Angles and Their Significance
Quadrantal angles are multiples of π/2 (90 degrees) that lie on the x- or y-axis of the unit circle. These angles divide the circle into four quadrants, each with specific signs for sine and cosine functions. Recognizing which quadrant an angle falls into helps determine the sign of trigonometric values.
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Sign of the Sine Function in Different Quadrants
The sine function corresponds to the y-coordinate on the unit circle. It is positive in the first and second quadrants (0 to π radians) and negative in the third and fourth quadrants (π to 2π radians). Identifying the quadrant of the angle allows you to decide if sin(θ) is positive or negative without a calculator.
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