Problem 18 - Algebra 2 and Trigonometry

Question

The coordinates of a point are given. a. Find the distance of the point from the origin. Express approximate distances to the nearest hundredth. b. Find the measure, to the nearest degree, of the angle in standard position whose terminal side contains the given point. \((12,-9)\)

Step-by-Step Solution

Answer

Distance is 15, and angle is approximately 323°.

Step 1: Understanding the Problem

We need to find the distance from the given point \((12, -9)\) to the origin and the angle formed by the line connecting the point to the origin with the positive x-axis, expressed in degrees.

Step 1: Calculating Distance

The distance formula is used to calculate the distance from the point \((x, y)\) to the origin \((0, 0)\). The formula is: \[ d = \sqrt{x^2 + y^2} \]. Substituting \(x = 12\) and \(y = -9\), we have \[ d = \sqrt{12^2 + (-9)^2} = \sqrt{144 + 81} = \sqrt{225} = 15 \].

Step 2: Calculating Angle in Radians

The angle in standard position can be found using the tangent function: \(\tan \theta = \frac{y}{x}\). For the point \((12, -9)\), \(\tan \theta = \frac{-9}{12} = -\frac{3}{4}\). The radians measure \(\theta = \arctan(-\frac{3}{4})\) can be calculated using a calculator.

Step 3: Converting Radians to Degrees

Once we have the measure in radians from a calculator, it needs to be converted into degrees. If \(\theta\) was found to be approximately \(-0.6435\) radians (which needs a calculator for precise value), multiply it by \(\frac{180}{\pi}\) to convert to degrees: \(-0.6435 \times \frac{180}{\pi} \approx -36.87^\circ \).

Step 4: Finding Angle in Correct Quadrant

Since the point \((12, -9)\) is in the fourth quadrant, the angle in standard position is measured clockwise from the positive x-axis. Hence, the angle is \(360^\circ - 36.87^\circ \approx 323.13^\circ\).

Key Concepts

Distance Formula

To find the distance between a point and the origin, you can use the distance formula. This helps in determining how far apart two points are in a coordinate plane. The formula you'll use is: \[ d = \sqrt{x^2 + y^2} \] where:

• \(x\) is the x-coordinate of the point,
• \(y\) is the y-coordinate of the point.

In the exercise, the point given is \((12, -9)\). By substituting these values into the formula, you calculate the distance to be \[ \sqrt{12^2 + (-9)^2} = \sqrt{144 + 81} = \sqrt{225} = 15 \].This means the point is 15 units away from the origin. Remember, the result of the distance formula is always a non-negative number, representing a physical distance.

Angle in Standard Position

An angle in standard position is measured from the positive x-axis, and its measure can be found using trigonometric functions like tangent. For a point \((x, y)\), the tangent function is given by:\[ \tan \theta = \frac{y}{x} \]In this case, the point is \((12, -9)\), so:\[ \tan \theta = \frac{-9}{12} = -\frac{3}{4} \]Using a calculator, you find:\[ \theta = \arctan\left(-\frac{3}{4}\right) \]This function helps us find the angle formed with respect to the x-axis in radians. The angle you find here is negative because it's in the fourth quadrant, where tangent values are negative.

Radians to Degrees Conversion

When you've got an angle in radians and need to express it in degrees, it's essential to remember that radians measure angles based on the radius of a circle, whereas degrees split a circle into 360 parts. To convert from radians to degrees, you use the formula:\[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]In the exercise, the angle in radians is approximately \(-0.6435\). By substituting into the conversion formula:\[ -0.6435 \times \frac{180}{\pi} \approx -36.87^\circ \]Because angles are usually reported in a positive degree measure from the positive x-axis, and the point is situated in the fourth quadrant, you calculate:\[ 360^\circ - 36.87^\circ \approx 323.13^\circ \]Thus, the angle in standard position with the positive x-axis is approximately \(323.13^\circ\). This conversion is crucial for interpreting angles in geometric problems accurately.