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Chapter 6: Problem 37

Find the angle \(\theta\) (in radians and degrees) between the lines. \(3 x+y=3\) \(x-y=2\)

Short Answer

Expert verified
The angle between the lines is \(\pi\)/4 radians or 45 degrees.

Step by step solution

01

Conversion to Slope-Intercept Form

Rewrite the given equations into the slope-intercept form (y = mx + b). The first equation becomes y= -3x + 3 and the second y = x - 2.
02

Calculate the Slopes

Identify the slopes of the lines. From the equations in step 1, the slope (m) of the first equation is -3 and that of the second equation is 1.
03

Use Slopes to Find Tangent of the Angle

Use the formula tan(鈭單�) = abs((m1 - m2) / (1 + m1*m2)) to find the tangent of the angle between the lines. This results in tan(鈭單�) = abs((-3 - 1) / (1 + -3*1)) = 1.
04

Calculate the Angle in Radians

Find the arctangent of the result from step 3 to get the angle in radians. 鈭單� = arctan(1) = \(\pi\)/4 or approximately 0.7854 radians.
05

Convert Angle to Degrees

Convert the angle from radians to degrees by multiplying by 180/\(\pi\). This gives 鈭單� = \(\pi\)/4 * (180/\(\pi\)) = 45 degrees.

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