Find cos 2A if tan A=5/12 and <A lies in quadrant III
I think I did this yesterday
5, 12, 13 triangle
cos^2 - sin^2
http://www.jiskha.com/display.cgi?id=1449008243
To find cos 2A, we can use the double angle identity for cosine:
cos 2A = cos^2 A - sin^2 A
Given that tan A = 5/12 and A lies in quadrant III, we can determine the values of sine and cosine for angle A.
In quadrant III, both sine and cosine are negative. Since tan A = sin A / cos A, we can determine the values of sine and cosine using the given tan A:
tan A = sin A / cos A
5/12 = sin A / (-cos A)
Let's solve this equation to find sin A:
cross-multiplying the equation, we have:
5*(cos A) = -12*(sin A)
5cos A = -12sin A
Simplifying, we get:
5cos A + 12sin A = 0
cos A = -12sin A / 5
Now, we can use the Pythagorean identity sin^2 A + cos^2 A = 1, to find the value of sin A:
(sin^2 A) + (-12sin A / 5)^2 = 1
25sin^2 A + 144sin^2 A = 25
169sin^2 A = 25
sin A = ±√(25/169)
sin A = ±(5/13)
Since A lies in quadrant III and sin A is negative in quadrant III, we have:
sin A = -5/13
Now that we have sin A and cos A, we can substitute these values into the double angle identity for cosine:
cos 2A = cos^2 A - sin^2 A
cos 2A = (-12sin A / 5)^2 - (-5/13)^2
cos 2A = (144(sin A)^2)/25 - 25/169
cos 2A = (144(-5/13)^2)/25 - 25/169
cos 2A = (144(25/169))/25 - 25/169
cos 2A = (12*25)/169 - 25/169
cos 2A = 300/169 - 25/169
cos 2A = (300-25)/169
cos 2A = 275/169
Therefore, cos 2A is equal to 275/169.
To find cos 2A, we can use the double angle formula for cosine:
cos 2A = cos^2 A - sin^2 A
Given that tan A = 5/12 and <A lies in quadrant III, we can determine the values of sin A and cos A as follows:
First, find the value of cos A:
cos A = 1 / sqrt(1 + tan^2 A)
= 1 / sqrt(1 + (5/12)^2)
= 1 / sqrt(1 + 25/144)
= 1 / sqrt(169/144)
= 1 / (13/12)
= 12/13
Since we know that <A lies in quadrant III (where cosine is negative), the value of cos A is -12/13.
Next, find the value of sin A using the Pythagorean identity:
sin A = sqrt(1 - cos^2 A)
= sqrt(1 - (-12/13)^2)
= sqrt(1 - (144/169))
= sqrt(25/169)
= 5/13
Now that we have sin A and cos A, we can plug these values back into the double angle formula to find cos 2A:
cos 2A = cos^2 A - sin^2 A
= (-12/13)^2 - (5/13)^2
= 144/169 - 25/169
= 119/169
Therefore, cos 2A is equal to 119/169.