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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 43
Capitolo 4, Problema 43

Without using a calculator, determine which of the two values is greater.
cos 2 or sin 2

Guida verificata passo dopo passo
1
Recall the fundamental trigonometric identity: \(\sin^2 x + \cos^2 x = 1\). This relationship connects sine and cosine values for the same angle.
Express \(\sin 2\) in terms of \(\cos 2\) using the identity. Since \(\sin^2 2 + \cos^2 2 = 1\), we have \(\sin 2 = \pm \sqrt{1 - \cos^2 2}\).
Determine the sign of \(\sin 2\) by considering the angle 2 radians. Since 2 radians is between \(\pi/2\) and \(\pi\) (approximately 1.57 to 3.14), \(\sin 2\) is positive in this interval.
Estimate the approximate values of \(\cos 2\) and \(\sin 2\) without a calculator by recalling the unit circle or known values: \(\cos 2\) is negative (since 2 radians is in the second quadrant), and \(\sin 2\) is positive.
Compare the two values based on their signs and approximate magnitudes: since \(\cos 2\) is negative and \(\sin 2\) is positive, conclude which is greater without calculating exact values.

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Understanding the Range and Behavior of Sine and Cosine Functions

Sine and cosine functions oscillate between -1 and 1. Knowing their values at specific angles, especially in radians, helps compare their magnitudes. Since 2 radians is between π/2 and π, sine and cosine have predictable signs and approximate values in this interval.
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Graph of Sine and Cosine Function

Evaluating Trigonometric Functions at Specific Angles Without a Calculator

Estimating sine and cosine values at non-standard angles involves understanding the unit circle and reference angles. For 2 radians, which is about 114.6°, sine is positive and cosine is negative, allowing qualitative comparison without exact calculation.
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Evaluate Composite Functions - Special Cases

Comparing Numerical Values of Trigonometric Functions

To determine which value is greater, compare the approximate magnitudes and signs of sine and cosine at the given angle. Since cosine 2 is negative and sine 2 is positive, sine 2 is greater, illustrating the importance of sign and magnitude in comparisons.
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6:04
Introduction to Trigonometric Functions
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