Ch. 1 - Angles and the Trigonometric Functions

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Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problema 1.RE.58

Capitolo 1, Problema 1.RE.58

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. sin (22𝜋/3)

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Recognize that the angle given is in radians: \(\frac{22\pi}{3}\). Since the sine function is periodic with period \(2\pi\), we can reduce the angle by subtracting multiples of \(2\pi\) to find a coterminal angle within the standard interval \([0, 2\pi)\).

Calculate how many full \(2\pi\) rotations fit into \(\frac{22\pi}{3}\). Since \(2\pi = \frac{6\pi}{3}\), divide \(\frac{22\pi}{3}\) by \(\frac{6\pi}{3}\) to find the quotient: \(\frac{22\pi/3}{6\pi/3} = \frac{22}{6} = 3 + \frac{4}{6} = 3 + \frac{2}{3}\).

Subtract \(3 \times 2\pi = 6\pi\) from \(\frac{22\pi}{3}\) to find the coterminal angle: \(\frac{22\pi}{3} - 6\pi = \frac{22\pi}{3} - \frac{18\pi}{3} = \frac{4\pi}{3}\).

Now, evaluate \(\sin \left( \frac{4\pi}{3} \right)\). Recall that \(\frac{4\pi}{3}\) is in the third quadrant where sine is negative, and it corresponds to an angle of \(\pi + \frac{\pi}{3}\).

Use the sine addition formula or reference the unit circle to express \(\sin \left( \frac{4\pi}{3} \right)\) as \(-\sin \left( \frac{\pi}{3} \right)\), and recall that \(\sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2}\).

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Radian Measure and Angle Conversion

Radians are a way to measure angles based on the radius of a circle, where 2π radians equal 360 degrees. Understanding how to convert between radians and degrees or simplify radian measures is essential for evaluating trigonometric functions accurately.

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Percorso guidato

5:04

Converting between Degrees & Radians

Periodic Properties of the Sine Function

The sine function is periodic with a period of 2π, meaning sin(θ) = sin(θ + 2πk) for any integer k. This property allows simplification of angles outside the standard interval [0, 2π) to equivalent angles within this range, facilitating exact value determination.

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Percorso guidato

5:33

Period of Sine and Cosine Functions

Exact Values of Sine for Special Angles

Certain angles, such as π/6, π/4, and π/3, have known exact sine values involving simple fractions and square roots. Decomposing or reducing complex angles into sums or differences of these special angles helps find exact sine values without a calculator.

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Percorso guidato

3:28

Common Trig Functions For 45-45-90 Triangles

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