At their point of intersection, the angle θbetween any two nonparallel lines satisfies the relat

Name: Problem 75, Chapter 6: Precalculus
Uploaded: 2020-05-31T18:18:05-08:00
Duration: 1 min 52 s
Channel: Andrew Bassila
Description: At their point of intersection, the angle θbetween any two nonparallel lines satisfies the relationship (m2-m1) cosθ= sinθ+m1 m2 sinθ, where m1 and m2 represent the slopes of the two lines. Rewrite the equation in terms of a single trig function.

step 1 All right, so this problem is talking about the angle between two non -parallel lines, and it gi

Question

At their point of intersection, the angle $\theta$ between any two nonparallel lines satisfies the relationship $\left(m_{2}-m_{1}\right) \cos \theta=$ $\sin \theta+m_{1} m_{2} \sin \theta,$ where $m_{1}$ and $m_{2}$ represent the slopes of the two lines. Rewrite the equation in terms of a single trig function.

   At their point of intersection, the angle $\theta$ between any two nonparallel lines satisfies the relationship $\left(m_{2}-m_{1}\right) \cos \theta=$ $\sin \theta+m_{1} m_{2} \sin \theta,$ where $m_{1}$ and $m_{2}$ represent the slopes of the two lines. Rewrite the equation in terms of a single trig function.

Precalculus

John W. Coburn 2nd Edition

Chapter 6, Problem 75

Instant Answer

Solved by Expert Andrew Bassila

05/31/2020

Step 1

We can see that $\sin \theta$ is common in the right side of the equation. So, we can factor out $\sin \theta$ from the right side of the equation. This gives us $\left(m_{2}-m_{1}\right) \cos \theta=$ $\sin \theta(1+m_{1} m_{2})$. Show more…

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At their point of intersection, the angle $\theta$ between any two nonparallel lines satisfies the relationship $\left(m_{2}-m_{1}\right) \cos \theta=$ $\sin \theta+m_{1} m_{2} \sin \theta,$ where $m_{1}$ and $m_{2}$ represent the slopes of the two lines. Rewrite the equation in terms of a single trig function.

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Key Concepts

Angle between lines in analytic geometry

This concept focuses on determining the measure of the angle at which two lines intersect, using their slopes. In analytic geometry, an important relationship exists between the slopes of intersecting lines and the tangent of the angle between them. Recognizing this relationship helps in simplifying expressions that involve trigonometric functions by connecting geometric interpretations with algebraic ones.

Rewriting trigonometric equations using identities

This key concept involves manipulating equations that contain multiple trigonometric functions to express them in terms of a single function. A common strategy is to divide the equation by one of the trigonometric functions (provided it is nonzero), which creates a tangent function, thereby consolidating sine and cosine into one function. This simplification process is widely used to solve and analyze trigonometric equations in various mathematical contexts.

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