Given that tanè = -5 and è is in the 2nd quadrant, compute exactly:
A. sin(è)
B. cos(è)
C. tan(2è)
The e is supposed to be theta.
In quad 2
sin is +
cos is -
of course we can look up the angle knowing the tangent and then find the sin etc but that is too easy.
tan T = -5 = sin T/cos T
cos T = sqrt (1 - sin^2 T)
so
-5 = sin T / sqrt(1-sin^2 T)
25 (1-sin^2 T ) = sin^2 T
25 = 26 sin^2 T
sin^2 T = 25/26
sin T = 5/sqrt(26)
then cos T = -sqrt (1 - 25/26)
now use
tan 2T = 2 tan T /(1-tan^2 T)
or
Just sketch the triangle in II
given tanè = -5/1 ----> y/x ---> y = 5, x = -1
r^2 = 5^2+1^1=26
r = √26
sinè = y/r = 5/√26
cosè = -1/√26
tan 2è = 2tanè/(1 - tan^2 è)
= -10/(1 - 25)
= -10/24
= 5/12
To find the values of sin(è), cos(è), and tan(2è), we need to use the given information that tan(è) = -5 and è is in the 2nd quadrant.
A. sin(è):
Since tan(è) is negative in the 2nd quadrant, we know that sin(è) must be positive. We can use the Pythagorean identity sin^2(è) + cos^2(è) = 1 to find the value of sin(è).
First, we can find cos(è):
Using the relationship between tangent and cosine, we have:
tan(è) = sin(è) / cos(è)
-5 = sin(è) / cos(è) [Substituting the given value of tan(è)]
We can rearrange this equation to solve for cos(è):
cos(è) = sin(è) / -5
Now, we can find sin(è):
Using the Pythagorean identity:
sin^2(è) + cos^2(è) = 1
sin^2(è) + (sin(è) / -5)^2 = 1 [Substituting the value of cos(è)]
simplifying the equation, we get:
sin^2(è) + sin^2(è) / 25 = 1
Combining the terms, we get:
(25*sin^2(è) + sin^2(è)) / 25 = 1
Multiplying through by 25, we get:
26*sin^2(è) = 25
Dividing through by 26, we get:
sin^2(è) = 25 / 26
Taking the square root of both sides, we get:
sin(è) = ±√(25 / 26)
But since è is in the 2nd quadrant (where sin is positive), we have:
sin(è) = √(25 / 26)
So, the value of sin(è) is √(25 / 26).
B. cos(è):
We previously found the value of cos(è) to be sin(è) / -5. Since we know sin(è) = √(25 / 26), we can substitute it into the equation:
cos(è) = √(25 / 26) / -5
So, the value of cos(è) is √(25 / 26) / -5.
C. tan(2è):
To find the value of tan(2è), we can use the double-angle identity for tangent:
tan(2è) = 2tan(è) / (1 - tan^2(è))
Substituting the given value of tan(è) = -5, we have:
tan(2è) = 2(-5) / (1 - (-5)^2)
Simplifying further, we get:
tan(2è) = -10 / (1 - 25)
tan(2è) = -10 / (-24)
Simplifying, we get:
tan(2è) = 5/12
So, the value of tan(2è) is 5/12.