Write each function as an expression involving functions of θ or x alone. See Example 2.
tan(180° + θ)

검증된 단계별 안내

Recall the angle addition formula for tangent: \(\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\).

Identify the angles in the expression: here, \(a = 180^\circ\) and \(b = \theta\).

Use the fact that \(\tan 180^\circ = 0\) because tangent is zero at \(180^\circ\).

Substitute into the formula: \(\tan(180^\circ + \theta) = \frac{\tan 180^\circ + \tan \theta}{1 - \tan 180^\circ \tan \theta} = \frac{0 + \tan \theta}{1 - 0 \cdot \tan \theta} = \tan \theta\).

Consider the sign of the tangent function in the third quadrant (where \(180^\circ + \theta\) lies) to determine the correct expression for \(\tan(180^\circ + \theta)\) in terms of \(\tan \theta\).

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주요 개념

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Angle Addition Formula for Tangent

The angle addition formula for tangent states that tan(a + b) = (tan a + tan b) / (1 - tan a tan b). This formula allows us to express the tangent of a sum of two angles in terms of the tangents of the individual angles, which is essential for rewriting tan(180° + θ).

Tangent Function Periodicity

The tangent function has a period of 180°, meaning tan(θ + 180°) = tan θ. This property simplifies expressions involving angles shifted by 180°, allowing us to rewrite tan(180° + θ) directly as tan θ.

Reference Angles and Quadrant Sign Rules

Understanding the signs of trigonometric functions in different quadrants helps determine the correct value of tan(180° + θ). Since 180° + θ lies in the third quadrant where tangent is positive, this confirms the sign of the expression after simplification.

Write each function as an expression involving functions of θ or x alone. See Example 2.tan(180° + θ)