Problem 41 - Basic Technical Mathematics with Calculus

Question

Draw angles in standard position such that the terminal side passes through the given point. $$(-7,5)$$

Step-by-Step Solution

Answer

The angle is approximately 144.46° in standard position.

Step 1: Understanding Standard Position

An angle is in standard position if its vertex is at the origin (0,0) and its initial side lies along the positive x-axis. The terminal side is determined by the point through which it passes.

Step 2: Identifying the Quadrant

The point (-7,5) indicates that you need to move 7 units to the left (negative x-direction) and 5 units up (positive y-direction). This places the point in Quadrant II.

Step 3: Calculating the Angle

Use the tangent function to find the angle's measure from the positive x-axis. The formula for the tangent of an angle θ is tan(θ) = y/x. For the point (-7,5), tan(θ) = 5/(-7).

Step 4: Finding the Reference Angle

Compute the reference angle by taking the arctan of the absolute value of tan(θ). This is θ_{ref} = an^{-1}(5/7).

Step 5: Converting to Standard Angle

In Quadrant II, the angle in standard position is θ = 180° - θ_{ref}. Use the reference angle calculated previously to find the standard position angle.

Key Concepts

Standard Position

When discussing angles in trigonometry, the term "standard position" refers to a specific orientation. An angle is considered to be in standard position if its vertex is located at the origin of a coordinate plane, specifically at (0,0), and its initial side is positioned along the positive x-axis. This orientation allows for consistent measurements and easy determination of other properties related to the angle. The other ray of the angle, known as the terminal side, determines the actual measure of the angle by extending through a particular point on the plane. It's important to understand that as the terminal side rotates counterclockwise from the positive x-axis, it describes an angle's measure that increases in the positive direction.
In our exercise, the terminal side of the angle passes through the point (-7,5). By identifying this point, we can further explore and calculate the properties and exact measures of the angles involved.

Quadrant Identification

A crucial step in working with angles in the coordinate plane is identifying the quadrant in which the terminal side lies. The plane is divided into four regions, called quadrants, by the x and y axes:

• Quadrant I: where both x and y coordinates are positive.
• Quadrant II: where the x-coordinate is negative, and the y-coordinate is positive.
• Quadrant III: where both x and y coordinates are negative.
• Quadrant IV: where the x-coordinate is positive, and the y-coordinate is negative.

In our scenario, the point (-7,5) shows a movement of 7 units left, meaning negative in the x-direction, and 5 units up, which is positive in the y-direction. These movements position the terminal side of the angle in Quadrant II. Knowing in which quadrant an angle's terminal side resides helps in determining other angle attributes, such as its tangent and reference angle.

Tangent Function

The tangent function plays a vital role in trigonometry and is closely related to the coordinate plane's angles. Given by \(\tan(\theta) = \frac{y}{x}\), the tangent of an angle helps us find the angle measure when a point on the terminal side is given. Here, \(y\) and \(x\) represent the coordinates of the point through which the terminal side passes.
For the point (-7,5), the calculation becomes \(\tan(\theta) = \frac{5}{-7}\).
This function indicates the ratio of the opposite side to the adjacent side on the coordinate grid. In Quadrant II, the tangent function will have a negative value due to the negative x-coordinate and positive y-coordinate. This negativity in the tangent value helps confirm the angle's placement in that quadrant.

Reference Angle

In trigonometry, the concept of the "reference angle" is essential for understanding angles that are positioned beyond the first quadrant. A reference angle is the acute angle formed by the terminal side of the given angle and the nearest x-axis. To find this angle, we use the inverse tangent function.
For our problem, the reference angle \(\theta_{\text{ref}}\) can be calculated by taking the arctan of the absolute value of \(\frac{5}{7}\).\[\theta_{\text{ref}} = \tan^{-1}\left(\frac{5}{7}\right)\]
This computation yields a positive acute angle in degrees or radians. When dealing with angles in standard position located in Quadrant II, the desired angle can be found by subtracting the reference angle from \(180^\circ\) or \(\pi\) radians. Thus, the formula \(\theta = 180^\circ - \theta_{\text{ref}}\) applies, ensuring that we account for the quadrant's range and location.