Answer :
To find the acute angle between the lines given by their equations, we will follow these steps:
### Step 1: Convert both equations to slope-intercept form
The equations provided are:
1. [tex]\( y - 3x + 120 = 0 \)[/tex]
2. [tex]\( y = 5x - 2 \)[/tex]
First, let's rewrite the first equation in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y - 3x + 120 = 0 \][/tex]
[tex]\[ y = 3x - 120 \][/tex]
The second equation is already in slope-intercept form:
[tex]\[ y = 5x - 2 \][/tex]
### Step 2: Identify the slopes of the lines
From their slope-intercept forms ([tex]\( y = mx + b \)[/tex]):
- The slope [tex]\( m_1 \)[/tex] of the first line [tex]\( y = 3x - 120 \)[/tex] is 3.
- The slope [tex]\( m_2 \)[/tex] of the second line [tex]\( y = 5x - 2 \)[/tex] is 5.
### Step 3: Calculate the tangent of the angle between the two lines
The formula to find the tangent of the angle [tex]\( \theta \)[/tex] between two lines with slopes [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] is:
[tex]\[ \tan(\theta) = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \][/tex]
Substitute the slopes [tex]\( m_1 = 3 \)[/tex] and [tex]\( m_2 = 5 \)[/tex]:
[tex]\[ \tan(\theta) = \left| \frac{5 - 3}{1 + 3 \cdot 5} \right| \][/tex]
[tex]\[ \tan(\theta) = \left| \frac{2}{1 + 15} \right| \][/tex]
[tex]\[ \tan(\theta) = \left| \frac{2}{16} \right| \][/tex]
[tex]\[ \tan(\theta) = \left| \frac{1}{8} \right| \][/tex]
[tex]\[ \tan(\theta) = 0.125 \][/tex]
### Step 4: Calculate the angle in radians
To determine the angle [tex]\( \theta \)[/tex] itself, we use the arctangent function:
[tex]\[ \theta = \arctan(0.125) \][/tex]
Thus, the angle in radians:
[tex]\[ \theta \approx 0.12435499454676144 \text{ radians} \][/tex]
### Step 5: Convert the angle to degrees
To convert radians to degrees, we multiply by [tex]\( \frac{180}{\pi} \)[/tex]:
[tex]\[ \theta \approx 0.12435499454676144 \times \frac{180}{\pi} \][/tex]
[tex]\[ \theta \approx 7.125016348901798^\circ \][/tex]
### Final Answer
The acute angle between the given pair of lines is approximately [tex]\( 7.125^\circ \)[/tex].
### Step 1: Convert both equations to slope-intercept form
The equations provided are:
1. [tex]\( y - 3x + 120 = 0 \)[/tex]
2. [tex]\( y = 5x - 2 \)[/tex]
First, let's rewrite the first equation in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y - 3x + 120 = 0 \][/tex]
[tex]\[ y = 3x - 120 \][/tex]
The second equation is already in slope-intercept form:
[tex]\[ y = 5x - 2 \][/tex]
### Step 2: Identify the slopes of the lines
From their slope-intercept forms ([tex]\( y = mx + b \)[/tex]):
- The slope [tex]\( m_1 \)[/tex] of the first line [tex]\( y = 3x - 120 \)[/tex] is 3.
- The slope [tex]\( m_2 \)[/tex] of the second line [tex]\( y = 5x - 2 \)[/tex] is 5.
### Step 3: Calculate the tangent of the angle between the two lines
The formula to find the tangent of the angle [tex]\( \theta \)[/tex] between two lines with slopes [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] is:
[tex]\[ \tan(\theta) = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \][/tex]
Substitute the slopes [tex]\( m_1 = 3 \)[/tex] and [tex]\( m_2 = 5 \)[/tex]:
[tex]\[ \tan(\theta) = \left| \frac{5 - 3}{1 + 3 \cdot 5} \right| \][/tex]
[tex]\[ \tan(\theta) = \left| \frac{2}{1 + 15} \right| \][/tex]
[tex]\[ \tan(\theta) = \left| \frac{2}{16} \right| \][/tex]
[tex]\[ \tan(\theta) = \left| \frac{1}{8} \right| \][/tex]
[tex]\[ \tan(\theta) = 0.125 \][/tex]
### Step 4: Calculate the angle in radians
To determine the angle [tex]\( \theta \)[/tex] itself, we use the arctangent function:
[tex]\[ \theta = \arctan(0.125) \][/tex]
Thus, the angle in radians:
[tex]\[ \theta \approx 0.12435499454676144 \text{ radians} \][/tex]
### Step 5: Convert the angle to degrees
To convert radians to degrees, we multiply by [tex]\( \frac{180}{\pi} \)[/tex]:
[tex]\[ \theta \approx 0.12435499454676144 \times \frac{180}{\pi} \][/tex]
[tex]\[ \theta \approx 7.125016348901798^\circ \][/tex]
### Final Answer
The acute angle between the given pair of lines is approximately [tex]\( 7.125^\circ \)[/tex].
