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Ch. 3 - Radian Measure and The Unit Circle
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 3 - Radian Measure and The Unit CircleProblema 56
Capitolo 4, Problema 56

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)

Guida verificata passo dopo passo
1
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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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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Percorso guidato
5:04
Converting between Degrees & Radians

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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Percorso guidato
6:36
Quadratic Formula

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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Percorso guidato
06:14
Sum and Difference of Sine & Cosine
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