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

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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