Identity for tan(x-y) = [tan(x) - tan(y)]/[1-tan(x)tan(y)]

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In summary, the conversation discusses the use of the identity for tan(x-y) and slopes of lines to show that the tangent of the angle between two intersecting lines can be expressed as m2-m1/(1 + m1m2). The conversation also clarifies that the slope of a line is the tangent of its inclination, not the angle itself.
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BrendanM
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Use the identity for tan(x-y) = [tan(x) - tan(y)]/[1-tan(x)tan(y)] to show that if two lines L1 and L2 intersect at angle theta then tan(theta) = m2-m1/(1 + m1m2) where m1 and m2 are the slopes of L1 and L2 respectivly.

hmm ihave no idea where to start for this it doesn't make sense to me. please help
 
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  • #2
Hint: The slope of a straight line is the angle the line makes with the x-axis.
 
  • #3
No, the slope of a line is the TANGENT of the angle the line makes with the x-axis. Which is the whole point of this exercise!

1) Draw a picture. You can always shift you coordinate system up or down, right or left without changing angles (or slopes) so draw it so that the two lines intersect at the origin. If the angles the two lines make with the x-axis are θ1 and θ2, what is the angle between them? What is the tangent of that angle?
 
  • #4
HallsofIvy is correct. The slope of a line is the tangent of its inclination.
 
  • #5
I knew that! Thanks for pointing out my typo! :-)
 

FAQ: Identity for tan(x-y) = [tan(x) - tan(y)]/[1-tan(x)tan(y)]

What is the identity for tan(x-y)?

The identity for tan(x-y) is [tan(x) - tan(y)]/[1-tan(x)tan(y)].

How is this identity derived?

This identity can be derived using the trigonometric identities for the sum and difference of two angles: tan(x+y) = (tan(x)+tan(y))/(1-tan(x)tan(y)) and tan(x-y) = (tan(x)-tan(y))/(1+tan(x)tan(y)). By substituting tan(x+y) into tan(x-y), the identity [tan(x) - tan(y)]/[1-tan(x)tan(y)] is obtained.

What is the purpose of this identity?

This identity is useful in simplifying complex trigonometric expressions involving the difference of two angles. It can also be used to solve trigonometric equations and prove other identities.

Can this identity be applied to other trigonometric functions?

Yes, this identity can also be applied to other trigonometric functions such as sine and cosine by using their respective identities for the sum and difference of two angles.

How can this identity be used in real-world applications?

This identity can be used in fields such as physics, engineering, and astronomy to calculate the difference in angles between two objects or to simplify equations involving trigonometric functions.

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