Find the exact value of: tan(-3pi).
very easy
remember that tanx = sinx/cosx
and that the sine curve has a zero for every multiple of 180º or every multiple of pi
Since -3pi is a multiple of pi, sin(-3pi) = 0
We know that the cosine cannot be zero at that same angle since the sine and cosine curves do not coincide
so tan(-3pi) = 0
To find the exact value of tan(-3π), we can use the periodicity property of tangents. The tangent function has a period of π, which means that the tangent of any angle is equal to the tangent of that angle plus or minus any multiple of π.
In this case, we have an angle of -3π. Since π is a multiple of π, we can add or subtract any multiple of π to get an equivalent angle. We can simplify -3π as -π - 2π.
Now, we can use the properties of tangents:
tan(-π - 2π) = tan(-π) = -tan(π)
The exact value of tan(π) is 0. Therefore, the exact value of tan(-3π) is -0 or simply 0.
To find the exact value of tan(-3π), we can use the periodicity property of the tangent function.
The tangent function has a period of π, which means that tan(x) = tan(x + π) for any value of x.
In this case, we can relate -3π to an equivalent angle within one period by adding 2π repeatedly until we find an angle within the range of -π to π.
Adding 2π to -3π, we get:
-3π + 2π = -π
Since tan(x) = tan(x + π), we can conclude that tan(-3π) is equal to tan(-π).
The tangent function is periodic with a period of π, meaning it has the same values at x and x + π. Therefore, tan(-3π) is equal to tan(-π).
Finally, we need to find the exact value of tan(-π).
The tangent function can be defined as the ratio of the sine and cosine functions: tan(x) = sin(x) / cos(x).
Since we have -π, we can find the exact values of sin(-π) and cos(-π) using the unit circle.
In the unit circle, the sine of an angle is the y-coordinate of the point on the unit circle that corresponds to that angle, and the cosine of an angle is the x-coordinate of that point.
For -π, the corresponding point on the unit circle is (-1, 0).
So, sin(-π) = -1 and cos(-π) = 0.
Now, we can substitute these values into the ratio:
tan(-π) = sin(-π) / cos(-π)
= (-1) / 0
However, division by zero is undefined in mathematics.
Therefore, we can conclude that the exact value of tan(-3π) is undefined.