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

Verified step by step guidance

1. 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.
2. 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°$.
3. Calculate the reduced angle: \(-1020° + 180° \times 6 = -1020° + 1080° = 60°\). So, \(\tan(-1020°) = \tan(60°)\).
4. Recall the exact value of \(\tan(60°)\) from the unit circle or special triangles. The tangent of 60° is \(\sqrt{3}\).
5. Therefore, the exact value of \(\tan(-1020°)\) is the same as \(\tan(60°)\), which is \(\sqrt{3}\).

Key Concepts

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.