In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. sin(-240°)

검증된 단계별 안내

Identify the given angle: \(-240^\circ\). Since it is negative, find its positive coterminal angle by adding \(360^\circ\): \(-240^\circ + 360^\circ = 120^\circ\).

Determine the reference angle for \(120^\circ\). Since \(120^\circ\) is in the second quadrant, the reference angle is \(180^\circ - 120^\circ = 60^\circ\).

Recall the sign of sine in the second quadrant. Sine is positive in the second quadrant, so \(\sin(120^\circ) = +\sin(60^\circ)\).

Use the exact value of \(\sin(60^\circ)\), which is \(\frac{\sqrt{3}}{2}\).

Therefore, \(\sin(-240^\circ) = \sin(120^\circ) = +\frac{\sqrt{3}}{2}\).

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주요 개념

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Reference Angles

A reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It helps simplify trigonometric calculations by relating any angle to a corresponding angle in the first quadrant, where values are easier to determine.

Trigonometric Function Signs in Quadrants

The sign of sine, cosine, and tangent functions depends on the quadrant in which the angle's terminal side lies. For sine, it is positive in the first and second quadrants and negative in the third and fourth quadrants, which is essential for determining the exact value.

Evaluating Sine of Negative Angles

The sine function is odd, meaning sin(-θ) = -sin(θ). This property allows converting negative angles into positive ones by changing the sign of the sine value, simplifying the evaluation without a calculator.