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**Monox D. I-Fly****Member**- From: Indonesia
- Registered: 2015-12-02
- Posts: 1,310

If 0 < x < 90, what is the solution of tan(2x - 5) = cot(x + 5)?

I got stuck in tan(2x - 5)tan(x + 5) = 1. What should I do after that?

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**Alg Num Theory****Member**- Registered: 2017-11-24
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Oops, that does not work. If tan*A*tan*B* = 1, then the denominator in the formula for tan(*A*+*B*) is 0.

Sorry, back to the drawing board …

*Last edited by Alg Num Theory (2018-01-29 03:45:41)*

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,442

try using:

cos(A-B) + cos(A+B) = 2 cos(A) . cos(B)

A = 2x-5 and B = x+5

and a similar formula with a minus sign.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob Bundy

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**Monox D. I-Fly****Member**- From: Indonesia
- Registered: 2015-12-02
- Posts: 1,310

What does it have to do with cos? The questions are tan and cot.

Actually I never watch Star Wars and not interested in it anyway, but I choose a Yoda card as my avatar in honor of our great friend bobbym who has passed away. May his adventurous soul rest in peace at heaven.

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**bob bundy****Administrator**- Registered: 2010-06-20
- Posts: 8,442

hi Monox D. I-Fly

Sorry my post was too brief. I was in a hurry as I had to go out. Here's some more detail:

note:

Let A = 2x-5 and B = x+5 => A-B = x - 10 and A+B = 3x

The two trig. formulas that will help are

and

so the equation becomes:

I'll leave you to finish this.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob Bundy

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**Alg Num Theory****Member**- Registered: 2017-11-24
- Posts: 339
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I have an idea: if tan*A*·tan*B* = 1, then *A*+*B* must be an odd multiple of 90°. Example:

and 30° + 60° = 90°.

In the OP’s example, 0 < x < 90°, so 0 < (2*x* + 5) + (*x* − 5) = 3*x* < 270° ⇒ *x* = 90°.

PS: Check: tan(175°) = 1/tan(95°) so I think I’m on the right track.

*Last edited by Alg Num Theory (2018-01-30 03:27:23)*

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**Monox D. I-Fly****Member**- From: Indonesia
- Registered: 2015-12-02
- Posts: 1,310

Huh? You got 90°? What I got was 30°.

Actually I never watch Star Wars and not interested in it anyway, but I choose a Yoda card as my avatar in honor of our great friend bobbym who has passed away. May his adventurous soul rest in peace at heaven.

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**Alg Num Theory****Member**- Registered: 2017-11-24
- Posts: 339
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Sorry, I meant to say 3*x* = 90°, so yes, *x* = 30°.

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