Find the angles between the following pairs of lines:
a) $3 x-4 y=3$ and $5 x+12 y=13$
Answer (2)
Rewrite the equations of the lines in slope-intercept form to find their slopes: m 1 = 4 3 and m 2 = − 12 5 .
Use the formula tan ( θ ) = ∣ 1 + m 1 m 2 m 1 − m 2 ∣ to find the tangent of the angle between the lines.
Calculate tan ( θ ) = 33 56 .
Find the angle θ by taking the arctangent: θ = arctan ( 33 56 ) ≈ 59.4 9 ∘ .
59.4 9 ∘
Explanation
Problem Analysis The problem asks us to find the angle between two given lines. The equations of the lines are 3 x − 4 y = 3 and 5 x + 12 y = 13 . To find the angle, we will first determine the slopes of the lines and then use the formula for the angle between two lines in terms of their slopes.
Finding the slopes First, we need to express each equation in the slope-intercept form, which is y = m x + c , where m is the slope and c is the y-intercept.
For the first line, 3 x − 4 y = 3 , we can rearrange it as follows: 4 y = 3 x − 3 y = 4 3 x − 4 3 So, the slope of the first line, m 1 , is 4 3 .
For the second line, 5 x + 12 y = 13 , we can rearrange it as follows: 12 y = − 5 x + 13 y = − 12 5 x + 12 13 So, the slope of the second line, m 2 , is − 12 5 .
Using the formula for the angle between two lines Now that we have the slopes of the two lines, we can use the formula to find the angle θ between them: tan ( θ ) = ∣ 1 + m 1 m 2 m 1 − m 2 ∣ Plugging in the values of m 1 and m 2 :
tan ( θ ) = ∣ 1 + ( 4 3 ) ( − 12 5 ) 4 3 − ( − 12 5 ) ∣ tan ( θ ) = ∣ 1 − 48 15 4 3 + 12 5 ∣ tan ( θ ) = ∣ 1 − 16 5 12 9 + 12 5 ∣ tan ( θ ) = ∣ 16 16 − 16 5 12 14 ∣ tan ( θ ) = ∣ 16 11 6 7 ∣ tan ( θ ) = ∣ 6 7 × 11 16 ∣ tan ( θ ) = ∣ 3 × 11 7 × 8 ∣ tan ( θ ) = 33 56
Calculating the angle To find the angle θ , we take the arctangent of 33 56 :
θ = arctan ( 33 56 ) θ ≈ 59.4 9 ∘
Final Answer The angle between the two lines is approximately 59.4 9 ∘ .
Examples
Understanding the angles between lines is crucial in many real-world applications. For example, architects use these calculations to design buildings, ensuring walls meet at precise angles for structural integrity and aesthetic appeal. Similarly, in navigation, pilots and sailors use angles between their paths and landmarks to determine their position and heading. In computer graphics, calculating angles between lines is essential for rendering 3D images and creating realistic visual effects.
To find the angle between the lines given by the equations 3 x − 4 y = 3 and 5 x + 12 y = 13 , we determined their slopes as m 1 = 4 3 and m 2 = − 12 5 . Using the tangent formula, we calculated the angle to be approximately 59.4 9 ∘ .
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