What is tan θ in this diagram?

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Discussion Overview

The discussion revolves around determining the value of tan θ in a given geometric diagram. Participants explore various mathematical approaches and relationships involving trigonometric functions, particularly focusing on the tangent function in relation to other angles and sides of triangles.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses difficulty in finding tan θ, having calculated cos θ using the cosine rule, but finds the expression complex and unhelpful.
  • Another participant suggests expressing tan θ in terms of known tangents of other angles.
  • A different participant proposes forming a right-angled triangle involving θ and the sides of the diagram, hinting at drawing a full rectangle to aid in the solution.
  • Further contributions involve using relationships between sine and tangent, with one participant providing a formula involving sin θ and sin α derived from the triangle's geometry.
  • Another participant questions the complexity of the approach and suggests using the tangent subtraction formula directly.
  • One participant notes that by inspecting the diagram, they can establish a relationship for tan(π/4 - θ) and proposes expanding it to solve for tan θ.
  • Several participants acknowledge that the cosine rule may not be necessary for the solution, indicating a potential simplification in the approach.
  • There is a discussion about the resulting equation after expansion, with one participant clarifying that they need to solve the equation rather than simply stating the result.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to find tan θ. Multiple competing views and methods are presented, and the discussion remains unresolved regarding the most efficient or correct solution.

Contextual Notes

Some participants reference specific trigonometric identities and relationships, but the discussion includes various assumptions and conditions that are not fully resolved. The complexity of the expressions and the reliance on geometric interpretations contribute to the ongoing debate.

Who May Find This Useful

This discussion may be useful for individuals interested in trigonometry, geometric interpretations of angles, and problem-solving strategies in mathematics.

PsychoMessiah
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a41lrt.png


< Moderator Note -- Thread moved from the technical math forums (that's why the HH Template is not shown) >[/color]

It's supposed to be a simple problem. But I can't for the life of me figure out how to go about it. I managed to find out cos θ using the cosine rule, but it is a very long expression and looks to be going in a direction opposite of the solution. cos θ is (2x^2 + 2xy + y^2 + x*sqrt(2) - y) / (2 * (2x^2 + 2xy + y^2) * (x*sqrt2)).

Any help on this would be appreciated.
 
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Can you express tan theta in terms of two other angles you know the tangents for?
 
Perhaps you can form a right-angled triangle involving theta and the two sides? Hint: try draw the full rectangle and see what that brings you. If you can't use angle theta directly then perhaps some other angle easily derived from it ...
 
PsychoMessiah said:
a41lrt.png


It's supposed to be a simple problem. But I can't for the life of me figure out how to go about it. I managed to find out cos θ using the cosine rule, but it is a very long expression and looks to be going in a direction opposite of the solution. cos θ is (2x^2 + 2xy + y^2 + x*sqrt(2) - y) / (2 * (2x^2 + 2xy + y^2) * (x*sqrt2)).

Any help on this would be appreciated.
From intersection point of pieces "x" and "y" put a normal "n" to a hypotenuze of a big triangle. Then you have:
n : x√2 = sin θ , n : y = sin α
From this you have: sin θ = (y⋅sin α)/(x√2)
Knowing that sin2α = x2/(x2+(x+y)2) and that 1+ctg2θ = 1/sin2θ , you should obtain correct result (A).
 
This does seem a bit long winded. Can you expand tan(a-b) directly in terms of tan a and tan b?
 
sjb-2812 said:
This does seem a bit long winded.

Untitledd9181.png

qrt{2}}\cdot%20\sin%20\alpha%20%3D\frac{y}{\sqrt{2}\cdot%20\sqrt{x^{2}&plus;%28x&plus;y%29^{2}}}.gif

^{2}%29}{y^{2}}-1%3D\frac{4x^{2}&plus;4xy&plus;y^{2}}{y^{2}}%3D\frac{%282x&plus;y%29^{2}}{y^{2}}.gif


1 min for drawing, 2 min for calculation, 3 min for Latex. This is how long it takes when derived from first principles.
 
By inspection of the diagram one can establish ##\tan(\pi/4-\theta) = x/(x+y)## from which it is easy to expand and solve for ##\tan(\theta)## (but here left as an exercise for the original poster).
 
Thanks for answering, everyone.

zoki85, that is very neatly done. Turns out we didn't need the cosine rule at all.

Filip Larsen, yes it is established that tan (45 - θ) = x / ( x + y ). But after expansion, we are left with (1 - tan θ) / (1 + tan θ ) using this formula...

idents07.gif


If you have an answer in mind, please share it.

Thanks.
 
PsychoMessiah said:
Thanks for answering, everyone.

zoki85, that is very neatly done. Turns out we didn't need the cosine rule at all.

Filip Larsen, yes it is established that tan (45 - θ) = x / ( x + y ). But after expansion, we are left with (1 - tan θ) / (1 + tan θ ) using this formula
We are not "left with" (1 - tan θ) / (1 + tan θ ) -- we are left with an equation whose right side is this. Write the whole equation and solve it for tan θ.
PsychoMessiah said:
idents07.gif


If you have an answer in mind, please share it.

Thanks.
 
  • #10
Right. Get it now. Thanks.
 

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