if tan(x/2)=senx and cosx is not zero, find tanx
a) -1 b) -(3^1/2)/3 c) 1/2 d) 1 e) 0
Please clarify your typo
is it
tan(x/2)=sinx
or
tan(x/2)=secx
btw, the bot's answer of tanx = 1, would give us 45°
then tan(22.5) = sin45 would be false, and
then tan(22.5) = sec 45 would also be false
so the robot tutor would be wrong in either case.
but tan(22.5) = .414213562..
sec(45) = 1.414213562... , interesting.
since tan(x/2) = sinx/(1+cosx), we have
sinx/(1+cosx) = sinx
so sinx=0 gives x=0
and tanx = 0
or
1/(1+cosx) = 1
1 = 1+cosx
but cosx≠0 so that's out
To find the value of tan(x), we need to first determine the value of x using the given equation.
We are given that tan(x/2) = sin(x) and that cos(x) is not zero.
Using the trigonometric identity for half angle, tan(x/2) = sin(x) / (1 + cos(x)).
Substituting this into the given equation, we have:
sin(x) / (1 + cos(x)) = sin(x).
To solve this equation, we can cross-multiply:
sin(x) = sin(x) * (1 + cos(x)).
Expanding the expression on the right side, we get:
sin(x) = sin(x) + sin(x) * cos(x).
Next, we can subtract sin(x) from both sides:
0 = sin(x) * cos(x).
Since cos(x) is not zero, we can divide both sides of the equation by cos(x):
0 = sin(x).
This shows that sin(x) is equal to 0.
Now, knowing that sin(x) = 0, we can determine the possible values of x.
The sine function evaluates to zero at certain angles. These angles include 0, π, 2π, etc., as well as their respective negatives (-π, -2π, etc.).
Among these possible values of x, we need to find the ones where cos(x) is not zero (as stated in the given condition).
Considering the options provided:
a) -1: cos(-1) is not zero
b) -(3^1/2)/3: cos((-(3^1/2)/3) * π) is not zero
c) 1/2: cos((1/2) * π) is zero, so it does not satisfy the given condition
d) 1: cos(1) is not zero
e) 0: cos(0) is not zero
Therefore, the possible values of x are -1 and 1, which means that tan(x) can be found using both values.
Therefore, the correct answer is:
tan(x) = -1 and 1.