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

Verified step by step guidance

1. Recall the fundamental trigonometric identity: \(\sin^2 x + \cos^2 x = 1\). This relationship connects sine and cosine values for the same angle.
2. Express \(\sin 2\) in terms of \(\cos 2\) using the identity. Since \(\sin^2 2 + \cos^2 2 = 1\), we have \(\sin 2 = \pm \sqrt{1 - \cos^2 2}\).
3. 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.
4. 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.
5. 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.

Key Concepts

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.

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.

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.