# What is tan θ in this diagram?

1. Nov 24, 2014

### PsychoMessiah

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

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.

Last edited by a moderator: Nov 25, 2014
2. Nov 24, 2014

### sjb-2812

Can you express tan theta in terms of two other angles you know the tangents for?

3. Nov 24, 2014

### Filip Larsen

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 ...

4. Nov 24, 2014

### zoki85

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).

5. Nov 25, 2014

### sjb-2812

This does seem a bit long winded. Can you expand tan(a-b) directly in terms of tan a and tan b?

6. Nov 25, 2014

### zoki85

1 min for drawing, 2 min for calculation, 3 min for Latex. This is how long it takes when derived from first principles.

7. Nov 25, 2014

### Filip Larsen

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).

8. Nov 25, 2014

### PsychoMessiah

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...

Thanks.

9. Nov 25, 2014

### Staff: Mentor

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.

10. Nov 26, 2014

### PsychoMessiah

Right. Get it now. Thanks.