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#1 2025-02-02 01:11:15
Trigonometry question!
Hi guys,
I have a trigonometry question that's causing me a headache. The question is:
## Prove that: tan(70) = 2tan(50) + tan(20)
I had to tear 18 pages for this question. Please help me.
"Talent hits the target no one else can hit. Genius hits the target no one else can see." - Arthur Schopenhauer
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#2 2025-02-02 06:23:24
- KerimF
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Re: Trigonometry question!
It is like... tan(65) = 2tan(45) + tan(25) [wrong] Thanks to Phrzby Phil
It is like... tan(65) = 2tan(40) + tan(25)
Last edited by KerimF (2025-02-02 09:21:07)
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#3 2025-02-02 08:56:42
- Phrzby Phil
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Re: Trigonometry question!
KerimF - You mean: It is like... tan(65) = 2tan(40) + tan(25)
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#4 2025-02-02 09:17:57
- KerimF
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Re: Trigonometry question!
The general form of the OP equation:
tan(A) = 2*tan(2*A-90) - tan(90-A) {equation 0}
Let us start with the identity:
tan(a-b) = [ tan(a) - tan(b) ] / [ 1 + tan(a)*tan(b) ]
By replacing:
a = A
b = 90-A
We get:
tan[A-(90-A)] = [ tan(A) - tan(90-A) ] / [ 1 + tan(A)*tan(90-A) ] {equation 1}
Now let us evaluate the product: tan(A)*tan(90-A)
By using the identity:
tan(a+b) = [ tan(a) + tan(b) ] / [ 1 - tan(a)*tan(b) ]
We get:
tan[A+(90-A)] = [ tan(A) + tan(90-A) ] / [ 1 - tan(A)*tan(90-A) ]
tan(90) = [ tan(A) + tan(90-A) ] / [ 1 - tan(A)*tan(90-A) ]
But tan(90) = ∞
∞ = [ tan(A) + tan(90-A) ] / [ 1 - tan(A)*tan(90-A) ] which means that the denominator of the fraction must be zero. {x/0 = ∞}
1 - tan(A)*tan(90-A) = 0
tan(A)*tan(90-A) = 1
After replacing tan(A)*tan(90-A) by 1 in {equation 1}, we get:
tan[A-(90-A)] = [ tan(A) - tan(90-A) ] / ( 1 + 1 )
tan[A-(90-A)] = [ tan(A) - tan(90-A) ] / 2
2*tan(2*A-90) = tan(A) - tan(90-A)
tan(A) = 2*tan(2*A-90) + tan(90-A) which is similar to {equation 0}
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#5 2025-02-02 09:19:17
- KerimF
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Re: Trigonometry question!
KerimF - You mean: It is like... tan(65) = 2tan(40) + tan(25)
Thank you.
Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).
But only a human may have the freedom and ability to oppose his natural robotic nature.
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#6 2025-02-02 10:07:43
- Phrzby Phil
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Re: Trigonometry question!
Whew. Nicely done.
I was never gonna get this one.
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#7 2025-02-02 10:51:36
- KerimF
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Re: Trigonometry question!
Whew. Nicely done.
I was never gonna get this one.
I believe you were able to get it, if you had enough free time.
Last edited by KerimF (2025-02-02 10:54:04)
Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).
But only a human may have the freedom and ability to oppose his natural robotic nature.
But, by opposing it, such a human becomes no more of this world.
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#8 2025-02-02 15:05:19
Re: Trigonometry question!
Hi guys,
Thanks for ur answers but the question I asked was correct.
There were two questions in that exercise.
@@Prove that:
## tan(70) = 2tan(50) + tan(20)
## 2tan(70) = tan(80) - tan(10)
These two questions are similar.
These are questions from Opt. Maths of 9th grade.
"Talent hits the target no one else can hit. Genius hits the target no one else can see." - Arthur Schopenhauer
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#9 2025-02-02 22:01:47
- KerimF
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Re: Trigonometry question!
Hi guys,
Thanks for ur answers but the question I asked was correct.
There were two questions in that exercise.
@@Prove that:
## tan(70) = 2tan(50) + tan(20)
## 2tan(70) = tan(80) - tan(10)These two questions are similar.
These are questions from Opt. Maths of 9th grade.
Of course, you asked a correct question.
We just liked to prove it in its general form:
## tan(70) = 2tan(50) + tan(20)
## tan(a) = 2tan(b) + tan(c)
The two sides are equal if'
a+c=90 and b=a-c
Similarly:
## 2tan(70) = tan(80) - tan(10)
## 2tan(b) = tan(a) - tan(c)
The two sides are equal if'
a+c=90 and b=a-c
The trick is that:
1 - tan(70)*tan(20) = 0
1 - tan(80)*tan(10) = 0
1 - tan(a)*tan(c) = 0 , if a+c = 90 , see the denominator of the identity below:
tan(a+c) = tan(90) = [ tan(a) + tan(c) ] / [ 1 - tan(a)*tan(c) ] = ∞ {a fraction equals infinity, if its denominator=0 and its denominator≠0}
Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).
But only a human may have the freedom and ability to oppose his natural robotic nature.
But, by opposing it, such a human becomes no more of this world.
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#10 2025-02-03 22:33:22
- Bob
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Re: Trigonometry question!
hi KerimF
Thanks for supplying a method. I did try playing around with trig identities but didn't get there.
Sorry to be pedantic but tan(90) isn't defined ('cause you cannot play around with infinity like it's a number). But your method can be easily adapted 
.
Firstly
for all A except 0 and ∞.
This follows from the basic definitions of trig functions.
In a right angled triangle tanx = O/A and tan(90-x) = A/O.
Trig functions where x is not an angle in a right angled triangle are defined by a rotating unit line which ensures that any identity that is true for acute angles also holds for other angles.
So start with
This cancels down to
From which your identity follows.
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
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#11 2025-02-04 02:13:54
- KerimF
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Re: Trigonometry question!
Hi Bob,
Your solution is better since it doesn't use trig identities. Thank you.
Kerim
Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).
But only a human may have the freedom and ability to oppose his natural robotic nature.
But, by opposing it, such a human becomes no more of this world.
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#12 2025-02-04 22:16:33
- Bob
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Re: Trigonometry question!
I would call
trig identities.
Def: identity --- a formula that is true for all values of the variable(s).
https://www.mathwords.com/i/identity.htm
I started with your proof, which works if you allow tan(90) = ∞ and tan(90).tan(0) = 1 .
I just juggled the algebra to avoid messing with ∞.
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
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#14 2025-02-05 00:30:12
- KerimF
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Re: Trigonometry question!
Thanks guys.
My teacher gave a solution. Would you like to see?
Why not? The beauty of math is that 'all roads lead to Rome' I mean the solution of a problem (Rome) could be reached by following different logical paths (well-done roads).
Last edited by KerimF (2025-02-05 00:35:26)
Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).
But only a human may have the freedom and ability to oppose his natural robotic nature.
But, by opposing it, such a human becomes no more of this world.
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#15 2025-02-07 00:32:55
Re: Trigonometry question!
Ok! He did:
## soln,
70 = 50 + 20
taking tan both sides
tan70 = tan(50 + 20)
tan70 = [tan50 + tan20] / [1 - tan50.tan20]
tan70 - tan70.tan50.tan20 = tan50 + tan20
tan70 - tan(90 - 20).tan50.tan20 = tan50 + tan20
tan70 - cot20.tan50.tan20 = tan50 + tan20
tan70 - tan50 = tan50 + tan20
tan70 = 2tan50 + tan20 $$ proved
## Isn't it cool?
And I tore 18 pages for it. 
"Talent hits the target no one else can hit. Genius hits the target no one else can see." - Arthur Schopenhauer
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#16 2025-02-07 08:29:50
- Bob
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Re: Trigonometry question!
Cool indeed! Thanks for posting.
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
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