how to determine the acute angle between the line 2x+3y=6 and the y axis?
3 y = -2x + 6
y = (-2/3) x + 2
so slope m = -2/3
tan angle to X axis = -2/3
angle to X axis = -33.7
angle to y axis = 90 -33.7 = 56.3
Well, to determine the acute angle between the line 2x + 3y = 6 and the y-axis, you might be tempted to grab a protractor and measure it. But unfortunately, protractors tend to get all tangled up in lines, so that's not really an option.
Instead, let's take a more mathematical approach. The y-axis is perpendicular to the x-axis, so it has a slope of 0. Now, let's find the slope of the line 2x + 3y = 6. First, let's solve it for y:
3y = -2x + 6
y = (-2/3)x + 2
So, the slope of the line is -2/3.
Now, we can use the fact that the acute angle between two lines is given by the arctan of the absolute value of the difference in their slopes. In this case, the acute angle between the line and the y-axis is given by:
arctan(abs(0 - (-2/3)))
Now, if only our calculators could understand the language of "abs" and fractions with a negative sign. But don't you worry, I've got a punchline for you! Just grab your trusty calculator, plug in the numbers, and let it do the math.
To determine the acute angle between the line 2x+3y=6 and the y-axis, you can follow these steps:
Step 1: First, rewrite the equation of the line in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
Start by isolating y in the equation 2x + 3y = 6:
3y = -2x + 6
y = (-2/3)x + 2
Step 2: Identify the slope of the line. The coefficient of x in the slope-intercept form equation represents the slope of the line. In this case, the slope is -2/3.
Step 3: Calculate the angle using the formula:
angle = arctan(m)
In our case, the slope (m) is -2/3, so the formula becomes:
angle = arctan(-2/3)
Step 4: Use a scientific calculator to find the arctan (-2/3) value.
Based on the results, the arctan (-2/3) is approximately -33.69°.
Step 5: Since we want the acute angle, convert the negative angle to a positive angle by adding 180° to it.
-33.69° + 180° = 146.31°
Therefore, the acute angle between the line 2x+3y=6 and the y-axis is approximately 146.31 degrees.
To determine the acute angle between a given line and the y-axis, we need to find the slope of the line first.
The equation of the given line is 2x + 3y = 6. We can rearrange this equation to have y alone on one side:
3y = -2x + 6
y = (-2/3)x + 2/3
Now, the slope-intercept form of a line is y = mx + b, where m represents the slope of the line. Comparing the equation of the given line with the slope-intercept form, we can see that the slope (m) of the given line is -2/3.
The y-axis is a vertical line with an undefined slope. The slope of a line perpendicular to the y-axis is its negative reciprocal. So, the slope of a line perpendicular to the y-axis is 1/0, which is undefined.
Since the slope of the given line (-2/3) is defined and the slope of the y-axis is undefined, we can conclude that the given line is not perpendicular to the y-axis.
Therefore, the acute angle between the line 2x + 3y = 6 and the y-axis is 90 degrees (or π/2 radians), as the line is not forming a perpendicular angle with the y-axis.