Solve each problem. See Examples 3 and 4. The figure to the right indicates that the equation of a line passing through the point (a, 0) and making an angle θ with the x-axis is y = (tan θ) (x - a). Find an equation of the line passing through the point (5, 0) that makes an angle of 15° with the x-axis.

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Identify the given point through which the line passes: (5, 0).

Recognize that the line makes an angle $ \theta = 15^\circ $ with the x-axis.

Recall the formula for the equation of a line passing through $ (a, 0) $ and making an angle $ \theta $ with the x-axis: $\n\[\n\]$ y = (\tan \theta)(x - a) $$

Substitute the given values $ a = 5 $ and $ \theta = 15^\circ $ into the formula: $\n\[\n\]$ y = (\tan 15^\circ)(x - 5) $$

This expression represents the equation of the line. To write it explicitly, you can leave it in this form or calculate $ \tan 15^\circ $ if needed.

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주요 개념

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Slope of a Line and Angle with the x-axis

The slope of a line is the tangent of the angle it makes with the positive x-axis. If a line forms an angle θ with the x-axis, its slope m = tan(θ). This relationship helps convert geometric angle information into algebraic slope form.

Point-Slope Form of a Line

The point-slope form expresses a line's equation using a known point (x₁, y₁) and slope m: y - y₁ = m(x - x₁). It is useful for writing the equation of a line when a point and slope are given, as in this problem.

Evaluating Trigonometric Functions at Specific Angles

To find the slope for a given angle, you must evaluate tan(θ) at that angle. For θ = 15°, tan(15°) is a specific value that can be found using a calculator or trigonometric identities, enabling precise slope calculation.