angle x lies in the third quadrant and tanx=7/24
determiner an exact value for cos2x
determiner an exact value for sin2x
sin ( 2 x ) = 2 tan x / ( 1 + tan² x )
cos ( 2 x ) = ( 1 - tan² x ) / ( 1 + tan² x )
tan x = 7 / 24 so:
cos ( 2 x ) = ( 1 - tan² x ) / ( 1 + tan² x )
cos ( 2 x ) = [ 1 - ( 7 / 24 )² ] / [ 1 + ( 7 / 24 )² ]
cos ( 2 x ) = ( 1 - 49 / 576) + ( 1 + 49 / 576 )
cos ( 2 x ) = ( 576 / 576 - 49 / 576 ) / ( 576 / 576 + 49 / 576 )
cos ( 2 x ) = ( 527 / 576 ) / (625 / 576 )
cos ( 2 x ) = 576 ∙ 527 / 576 ∙ 625
cos ( 2 x ) = 527 / 625
sin ( 2 x ) = 2 tan x / ( 1 + tan² x )
sin ( 2 x ) = 2 ∙ ( 7 / 24 ) / [ 1 + ( 7 / 24 )² ]
sin ( 2 x ) = ( 14 / 24 ) / ( 1 + 49 / 576 )
sin ( 2 x ) = ( 14 / 24 ) / ( 576 / 576 + 49 / 576 )
sin ( 2 x ) = ( 14 / 24 ) / ( 625 / 576 )
sin ( 2 x ) = 14 ∙ 576 / 24 ∙ 625
sin ( 2 x ) = 8064 / 15000
sin ( 2 x ) = 24 ∙ 336 / 24 ∙ 625
sin ( 2 x ) = 336 / 625
Well, well, well, looks like we've got ourselves an angle in the third quadrant. Get ready for some math comedy!
To find the exact value of cos(2x), we'll need to use some trigonometric identities and a little bit of clownish magic. Let's get started, shall we?
We know that tan(x) = 7/24. In the third quadrant, both cosine and sine are negative. So, let's first find the values of sine and cosine individually.
Since tan(x) = 7/24, we can use the Pythagorean identity to find the hypotenuse. The Pythagorean identity states that (sin(x))^2 + (cos(x))^2 = 1.
Now, since sine and cosine are both negative in the third quadrant, let's assign their values as sin(x) = -7/25 and cos(x) = -24/25.
Now, to find cos(2x), we can use the identity cos(2x) = cos^2(x) - sin^2(x). Substituting our values in, we get:
cos(2x) = (-24/25)^2 - (-7/25)^2
cos(2x) = 576/625 - 49/625
cos(2x) = (576 - 49)/625
cos(2x) = 527/625
And there you have it, the exact value of cos(2x) is 527/625. Ta da!
Now, let's move on to sin(2x). We can use the identity sin(2x) = 2*sin(x)*cos(x). Substituting our values in, we get:
sin(2x) = 2*(-7/25)*(-24/25)
sin(2x) = (2*7*24)/(25*25)
sin(2x) = 336/625
And voila, the exact value of sin(2x) is 336/625. Shall we move on to our next math circus act?
To determine the exact value of cos(2x), we need to use the double angle identity for cosine.
The double angle identity for cosine is:
cos(2x) = cos^2(x) - sin^2(x)
Step 1: Determine the value of cos(x) using the given information.
Since tan(x) = 7/24, we know that tan(x) = sin(x) / cos(x).
So, sin(x) = 7, and cos(x) = 24.
Step 2: Substitute the values of sin(x) and cos(x) into the double angle identity.
cos(2x) = cos^2(x) - sin^2(x)
cos(2x) = (24)^2 - (7)^2
cos(2x) = 576 - 49
cos(2x) = 527
Therefore, the exact value of cos(2x) is 527.
To determine the exact value of sin(2x), we can use the double angle identity for sine.
The double angle identity for sine is:
sin(2x) = 2sin(x)cos(x)
Step 1: Determine the value of sin(x) and cos(x) using the given information.
sin(x) = 7
cos(x) = 24
Step 2: Substitute the values of sin(x) and cos(x) into the double angle identity.
sin(2x) = 2sin(x)cos(x)
sin(2x) = 2(7)(24)
sin(2x) = 336
Therefore, the exact value of sin(2x) is 336.
To determine the exact value of cos2x and sin2x, we can use the identities involving tangent and reciprocal trigonometric functions.
Given that tanx = 7/24, we can determine the values of sinx and cosx using the reciprocal identities. In the third quadrant, where x lies, tangents are positive, and both sinx and cosx are negative.
Let's find sinx and cosx:
sinx = -√(1 - cos^2x)
cosx = -√(1 - sin^2x)
Using the given value of tanx = 7/24, we can find the values of sinx and cosx.
tanx = sinx/cosx = 7/24
Rearranging the equation:
sinx = (7/24)cosx
Squaring both sides:
sin^2x = (49/576)cos^2x
Using the identity sin^2x + cos^2x = 1:
(49/576)cos^2x + cos^2x = 1
Simplifying the equation:
(625/576)cos^2x = 1
cos^2x = 576/625
cosx = ±√(576/625) = ±(24/25) = -24/25 (since cosx is negative in the third quadrant)
Now, we have the value of cosx as -24/25.
To find cos2x and sin2x, we can use the double-angle identities:
cos2x = cos^2x - sin^2x
sin2x = 2sinxcosx
Substituting the values we know:
cos2x = (-24/25)^2 - [(49/576)(-24/25)^2]
sin2x = 2(7/24)(-24/25)
Simplifying:
cos2x = 576/625 - (576/625)(49/576)
sin2x = -2(7/25)
cos2x = 252/625
sin2x = -14/25
Therefore, the exact values for cos2x and sin2x are:
cos2x = 252/625
sin2x = -14/25