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.

Verified step by step guidance

1. Identify the given point through which the line passes: (5, 0).
2. Recognize that the angle θ the line makes with the x-axis is 15°.
3. Use the formula for the equation of a line passing through a point (a, 0) and making an angle θ with the x-axis: y = (\tan \theta) (x - a).
4. Substitute the given point (5, 0) into the formula, setting a = 5.
5. Substitute the angle θ = 15° into the formula, using \tan 15° for the slope.

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness, typically represented as 'm' in the slope-intercept form of a line equation, y = mx + b. In this context, the slope can be calculated using the tangent of the angle θ that the line makes with the x-axis, where m = tan(θ). This relationship is crucial for determining the direction of the line based on the given angle.

Point-Slope Form of a Line

The point-slope form of a line is expressed as y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. This form is particularly useful when you know a specific point through which the line passes and the slope. In the given problem, the point (5, 0) serves as (x₁, y₁), allowing us to apply this formula to find the equation of the line.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. In this problem, the tangent function is used to find the slope of the line based on the angle θ. Understanding how to apply these functions is essential for solving problems involving angles and their corresponding slopes in coordinate geometry.