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Hate Number Theory

Member

Registered: 2015-12-22

Posts: 11

3 Trig Problems

1. In $\triangle ABC$, we have $AB = 16$, $BC = 14$, and $\angle A = 60^\circ$. Find the sum of all possible values of $AC$.
2. In $\triangle ABC$, we have $AB = 14$, $BC = 16$, and $\angle A = 60^\circ$. Find the sum of all possible values of $AC$.
3. Let

be positive real numbers. Prove that \[\sqrt{a^2 - ab + b^2} + \sqrt{a^2 - ac + c^2} \ge \sqrt{b^2 + bc + c^2}.\]
Under what conditions does equality occur? That is, for what values of $a$, $b$, and $c$ are the two sides equal?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: 3 Trig Problems

hi  Hate Number Theory

Let's have a go at number 1 first.

There are two possible places for C, I've marked them C and C'.

The line BD bisects CC' at right angles.

AC + AC' = 2.AD, so you can use trig to get AD and hence answer the question.

Please have a go and post back your answer.  If that is OK we can progress to the others. 

Bob

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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

Relentless

Member

Registered: 2015-12-15

Posts: 631

Re: 3 Trig Problems

Hi, I would just use the cosine rule for the first two.

For problem one, say c=14 and b=16. For problem two, say c=16 and b=14.

Bob is right that there are two solutions for problem one. For problem two there are two solutions algebraically, but because one of them is negative there is actually only one answer.

Relentless

Member

Registered: 2015-12-15

Posts: 631

Re: 3 Trig Problems

For problem three, I think the algebra is extremely tedious to do the old-fashioned way. I have the computed answer with no working if that is all you wanted (I'm not sure it is).

Last edited by Relentless (2016-01-26 04:54:18)

Nehushtan

Member

Registered: 2013-03-09

Posts: 957

Re: 3 Trig Problems

1. In

, we have

,

, and

. Find the sum of all possible values of

.

2. In

, we have

,

, and

. Find the sum of all possible values of

.

3. Let

be positive real numbers. Prove that[list=*]
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Under what conditions does equality occur? That is, for what values of

,

, and

are the two sides equal?

3. Rewrite

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Let

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and consider the complex numbers

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[*]

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Then the given inequality follows from the triangle inequality for complex numbers:

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[*]

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Equality occurs if and only if the two complex numbers and the origin are collinear with the origin not in between, i.e. if and only if

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[/list]

Last edited by Nehushtan (2016-01-27 00:58:16)

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