Question on Tangent in Acute quadrant

Feb 2009
155
1
Hello everyone

Greetings. Need help on this textbook question:

The tangent for an acute angle x, is m. Find cos2x.
Texbook answer : (1-m^2)/(1+m^2)

My answer:
cos2x = cos^2 - sin^2
= 1 - (sin^2 /cos^2 )
= 1 - tan^2
= 1 - m^2

Appreciate your help to clarify, thank you.

Best regards
 

Plato

MHF Helper
Aug 2006
22,491
8,653
The tangent for an acute angle x, is m. Find cos2x.
Texbook answer : (1-m^2)/(1+m^2)
If $\tan(x)=m$ for $x\in I$ then $\cos(x)=\dfrac{1}{\sqrt{1+m^2}}~\&~\sin(x)=\dfrac{m}{\sqrt{1+m^2}}$
 
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Feb 2009
155
1
Dear Plato

Thank you for the response, your answer is correct.

Best regards
 
Last edited:
Feb 2009
155
1
Dear Plato

m, tan x = m/1. Triangle Law, cos x = 1/(1+m^2)^1/2

cos 2x = 2 cos^2x - 1
= 2/(m^2 + 1) - 1
= (1 - m^2)/(m^2 + 1)

Best regards
 

Plato

MHF Helper
Aug 2006
22,491
8,653
Dear Plato

m, tan x = m/1. Triangle Law, cos x = 1/(1+m^2)^1/2

cos 2x = 2 cos^2x - 1
= 2/(m^2 + 1) - 1
= (1 - m^2)/(m^2 + 1)
In a right triangle with an acute angle measuring $x$ such that $\tan(x)=\frac{m}{1}$, then the hypotenuse measures $\sqrt{1+m^2}$

From those numbers find $\cos(x)~\&~\sin(x)~!$
 
Feb 2009
155
1
Hi Plato

That is the working for the final answer. Look at the question before you comment.

The tangent for an acute angle x, is m. Find cos2x.
Textbook answer : (1-m^2)/(1+m^2)


Best regards
 

Plato

MHF Helper
Aug 2006
22,491
8,653
Dear Plato

m, tan x = m/1. Triangle Law, cos x = 1/(1+m^2)^1/2

cos 2x = 2 cos^2x - 1
= 2/(m^2 + 1) - 1
= (1 - m^2)/(m^2 + 1)
In a right triangle with an acute angle measuring $x$ such that $\tan(x)=\frac{m}{1}$, then the hypotenuse measures $\sqrt{1+m^2}$

From those numbers find $\cos(x)~\&~\sin(x)~!$
 
Feb 2009
155
1
In a right triangle with an acute angle measuring $x$ such that $\tan(x)=\frac{m}{1}$, then the hypotenuse measures $\sqrt{1+m^2}$

From those numbers find $\cos(x)~\&~\sin(x)~!$
m, tan x = m/1. Triangle Law, cos x = 1/(1+m^2)^1/2

cos 2x = 2 cos^2x - 1
= 2/(m^2 + 1) - 1
= (1 - m^2)/(m^2 + 1

Hopefully the colored fonts will help you understand better.

Best regards
 
Last edited: