Find the exact value of the trigonometric function. -cot(pi/4 + 16pi)
16π is the same as exactly 8 rotations, so
-cot(π/4 + 16π) = -cot(π/4)
= -1/tan(π/4) = -1/1 = -1
cot ( A + B ) = 1 / tan ( A + B )
tan ( A + B )= ( tanA + tanB )/( 1 - tanA * tanB )
In this case:
A = pi / 4
B = 16 pi
tan ( pi / 4 ) = tan 45° = 1
tan ( 16 pi )= cot ( 8 * 2 pi ) = tan ( 8 * 360° ) = 0
tan ( pi / 4 + 16 pi ) =
[ ( tan ( pi / 4 ) + tan ( 16 pi ) ]/[ 1 - tan( pi / 4 ) * tan ( 16 pi ) ] =
( 1 + 0 ) / ( 1 - 1 * 0 ) =
1 / ( 1 - 0 ) =
1 / 1 = 1
tan ( pi / 4 + 16 pi ) = 1
cot ( pi / 4 + 16 pi ) =
1 / tan ( pi / 4 + 16 pi ) =
1 / 1 = 1
- cot ( pi / 4 + 16 pi ) = - 1
tan ( 16 pi )= tan ( 8 * 2 pi ) = tan ( 8 * 360° ) = 0
To find the exact value of the trigonometric function -cot(pi/4 + 16pi), we need to use the properties and definitions of trigonometric functions.
The cotangent function is defined as the reciprocal of the tangent function:
cot(x) = 1 / tan(x)
From the angle addition formula, we know that:
tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a) * tan(b))
In this case, a = pi/4 and b = 16pi. Substituting these values into the formula, we get:
tan(pi/4 + 16pi) = (tan(pi/4) + tan(16pi)) / (1 - tan(pi/4) * tan(16pi))
Now, let's find the values of tan(pi/4) and tan(16pi):
The value of tan(pi/4) is 1, because in the unit circle, the point (1, 1) lies on the line forming an angle of pi/4 radians with the positive x-axis.
The value of tan(16pi) can be determined by considering that the tangent function has a period of pi, meaning its values repeat every pi radians. Therefore, tan(16pi) is equivalent to tan(0), which is 0.
Substituting these values, we have:
tan(pi/4 + 16pi) = (1 + 0) / (1 - 1 * 0)
tan(pi/4 + 16pi) = 1 / 1
tan(pi/4 + 16pi) = 1
Now we can find the value of -cot(pi/4 + 16pi) by taking the reciprocal of the tangent value:
-cot(pi/4 + 16pi) = -1
Therefore, the exact value of the trigonometric function -cot(pi/4 + 16pi) is -1.