Problem 39 - Basic Technical Mathematics with Calculus

Question

Draw angles in standard position such that the terminal side passes through the given point. $$(-3,-5)$$

Step-by-Step Solution

Answer

The angle in standard position is approximately 239.04°.

Step 1: Understand the Standard Position

An angle is said to be in standard position if its vertex is at the origin of the coordinate plane, and its initial side lies along the positive x-axis. We need to find the angle whose terminal side passes through the point (-3, -5).

Step 2: Plot the Point

Plot the point (-3, -5) on the coordinate plane. The point is in the third quadrant since both x and y coordinates are negative. The angle will be measured from the positive x-axis to this point.

Step 3: Compute the Reference Angle

To find the angle, we first compute the reference angle using the tangent function, since the tangent of the angle is the ratio of the y-coordinate to the x-coordinate. \[ \text{tan} \theta = \frac{y}{x} = \frac{-5}{-3} = \frac{5}{3} \] Calculate the reference angle \( \theta_r \) using the inverse tangent function: \[ \theta_r = \arctan\left(\frac{5}{3}\right) \]

Step 4: Determine the Angle in Standard Position

Since the point is in the third quadrant, the angle in standard position \( \Theta \) is given by adding 180° to the reference angle \( \theta_r \): \[ \Theta = 180° + \theta_r \] Use a calculator to find \( \theta_r \) and then compute \( \Theta \).

Step 5: Calculate the Final Angle value

Use a calculator to find the approximate value for the inverse tan: \[ \theta_r \approx 59.04° \] Then, calculate the angle in standard position: \[ \Theta = 180° + 59.04° = 239.04° \] The angle in standard position is approximately 239.04°.

Key Concepts

Standard Position

In geometry and trigonometry, an angle is said to be in "standard position" when its vertex is at the origin of the coordinate plane, and its initial side is along the positive x-axis. This is a crucial starting point for measuring angles from the positive x-axis, moving counter-clockwise.

This foundational perspective allows us to consistently define and calculate various angles on the coordinate plane. By maintaining a standard position, angles can be compared, and calculations for transformations and rotations can be handled reliably.

Reference Angle

A reference angle is a useful concept when working with angles not in the first quadrant. It is defined as the acute angle formed by the terminal side of an angle and the horizontal axis.

Reference angles are always measured as a positive acute angle (less than 90°) and are instrumental in determining the trigonometric ratios for angles in all quadrants. This is due to their property of having the same absolute trigonometric values as the angle itself, only differing by sign depending on the quadrant in which the angle resides.

• For angles in the third quadrant, as with this exercise, the reference angle is subtracted from or added to 180° to find the standard angle.
• Understanding reference angles helps simplify solving problems related to trigonometric functions by focusing on their basic acute angle counterparts.

Inverse Tangent

The inverse tangent, or arctangent (often written as \( \arctan \))), is a fundamental operation used to determine an angle from its tangent value.

In the context of trigonometry, tangent is the ratio of the opposite side to the adjacent side in a right triangle, or the ratio of the y-coordinate to the x-coordinate on the coordinate plane.

• For instance, given a ratio of \( \frac{5}{3} \), the inverse tangent provides us with an angle whose tangent is \( \frac{5}{3} \).
• Using a calculator, \( \arctan\left(\frac{5}{3}\right) \) yields approximately \( 59.04° \).

Inverse functions like this are critical for converting between angle measures and side ratios in trigonometric calculations.

Coordinate Plane

The coordinate plane is an essential tool in mathematics that fuses algebraic and geometric ideas to address spatial questions.A two-dimensional surface defined by a pair of perpendicular axes: the x-axis (horizontal) and the y-axis (vertical)

allows for the precise plotting and manipulation of geometric figures through algebraic calculations.

• Every point on the plane is described by an ordered pair \( (x, y) \), where 'x' denotes its horizontal position, and 'y' its vertical position.
• Plotting the point \( (-3, -5) \) in this workout shows it rests in the third quadrant, illustrating how negative x and y coordinates pair to indicate the plane's different sections.

This plane serves as the backdrop for many calculations in mathematics and forms the basis for determining angles, distances, and analysis across both pure and applied math fields.

Third Quadrant

The coordinate plane is divided into four quadrants, each defined by the sign and orientation of the x and y values in that section. The third quadrant is where both the x and y values are negative.

This section of the plane is pivotal when considering real-world contexts like physics or engineering, or when working through geometric or algebraic equations involving negative values.

• Angles in the third quadrant range between 180° to 270°.
• Understanding that this quadrant houses points with negative coordinate values aids in visualizing and analyzing positions and angles.

Grasping the location and properties of the third quadrant is essential for interpreting various geometric and trigonometric principles.