Find the exact value of each expression. See Example 3. tan(-1020°)

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Recognize that the tangent function has a period of 180°, meaning that $\tan(\theta) = \tan(\theta + 180°k)$ for any integer $k$. This allows us to reduce the angle to an equivalent angle between 0° and 180° (or between -90° and 90°) to simplify the calculation.

Start by adding or subtracting multiples of 180° to the angle $-1020°$ to find a coterminal angle within the standard range. For example, add $180° \times k$ where $k$ is chosen so that the resulting angle lies between $-180°$ and \$180°$.

Calculate the reduced angle: $-1020° + 180° \times 6 = -1020° + 1080° = 60°$. So, $\tan(-1020°) = \tan(60°)$.

Recall the exact value of $\tan(60°)$ from the unit circle or special triangles. The tangent of 60° is $\sqrt{3}$.

Therefore, the exact value of $\tan(-1020°)$ is the same as $\tan(60°)$, which is $\sqrt{3}$.

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주요 개념

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Angle Coterminality

Angles that differ by full rotations (multiples of 360°) share the same terminal side and thus have the same trigonometric values. To simplify an angle like -1020°, add or subtract 360° repeatedly until the angle lies within a standard range, typically 0° to 360° or -360° to 360°.

Tangent Function Periodicity

The tangent function has a period of 180°, meaning tan(θ) = tan(θ + 180°). This property allows further simplification of angles by reducing them modulo 180°, making it easier to find exact values for tangent expressions.

Exact Values of Tangent for Special Angles

Certain angles, such as 0°, 30°, 45°, 60°, and 90°, have known exact tangent values derived from the unit circle or special triangles. Recognizing these angles after simplification helps in determining the exact value of the tangent expression without a calculator.