In Exercises 49–59, find the exact value of each expression. Do not use a calculator. tan 120°

검증된 단계별 안내

Recall that the tangent function has a period of 180°, so \( \tan(120^\circ) = \tan(120^\circ - 180^\circ) = \tan(-60^\circ) \), but it is often easier to use the reference angle in the second quadrant directly.

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

Recall the sign of tangent in the second quadrant. Tangent is negative in the second quadrant because sine is positive and cosine is negative, and \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

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

Combine the sign and the reference angle value to write \( \tan(120^\circ) = -\sqrt{3} \).

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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 the given angle and the x-axis. For angles greater than 90°, finding the reference angle helps determine the trigonometric function's value by relating it to a known acute angle.

Signs of Trigonometric Functions in Quadrants

The sign of trigonometric functions depends on the quadrant in which the angle lies. For example, tangent is positive in the first and third quadrants and negative in the second and fourth. Knowing the quadrant of 120° helps determine the sign of tan 120°.

Exact Values of Trigonometric Functions for Special Angles

Certain angles like 30°, 45°, 60° have known exact trigonometric values expressed in simplified radical form. Using these known values for the reference angle allows calculation of the exact value of tan 120° without a calculator.