find the exact value of sin(-15pi/6) and cot(-7pi/6)
Thank youu(:
To find the exact value of sin(-15π/6) and cot(-7π/6), we can use the unit circle and the periodicity of trigonometric functions.
1. For sin(-15π/6):
The reference angle for -15π/6 is π/6.
On the unit circle, at π/6, the y-coordinate is 1/2.
Since sin(-θ) = -sin(θ), the value of sin(-15π/6) is -1/2.
2. For cot(-7π/6):
The reference angle for -7π/6 is π/6.
On the unit circle, at π/6, the x-coordinate is √3/2 while the y-coordinate is -1/2.
cot(θ) = cos(θ) / sin(θ)
Therefore, cot(-7π/6) = cos(-7π/6) / sin(-7π/6)
= (√3/2) / (-1/2)
= -√3
So, the exact value of sin(-15π/6) is -1/2 and the exact value of cot(-7π/6) is -√3.
To find the exact value of sin(-15π/6) and cot(-7π/6), we can use the unit circle and the trigonometric identities.
1) Finding the exact value of sin(-15π/6):
Since we know that the sine function is periodic with a period of 2π, we can add or subtract any integer multiple of 2π to the given angle without changing its sine value. In this case, -15π/6 is equivalent to -π/2.
On the unit circle, the y-coordinate represents the sine value. At -π/2 angle, the y-coordinate is -1. Therefore, the exact value of sin(-15π/6) is -1.
2) Finding the exact value of cot(-7π/6):
The cotangent function is defined as the reciprocal of the tangent function, so cot(x) = 1/tan(x).
For -7π/6 angle, we can convert it to the equivalent angle within one full revolution. Adding or subtracting a multiple of 2π doesn't change the cotangent value. So we can add 2π to -7π/6 to get an equivalent angle.
-7π/6 + 2π = 5π/6
Now, we can find the tangent of 5π/6. On the unit circle, the tangent of an angle is equal to the y-coordinate divided by the x-coordinate.
At 5π/6 angle, the coordinates are: (x, y) = (-√3/2, 1/2)
Therefore, the tangent of 5π/6 is y/x = (1/2) / (-√3/2) = -1/√3 = -√3/3
Since we are looking for the cotangent, which is the reciprocal of the tangent, we can take the reciprocal of -√3/3:
cot(5π/6) = 1 / (-√3/3) = -√3
So the exact value of cot(-7π/6) is -√3.
Perhaps you can "think" better in degrees than radians, sometimes I can.
sin(-15π/6)
= sin(-450°)
= sin(-90) , I took away one rotation
= -sin(90°) = -1
cot (-7π/6)
= cot(-210°)
= cot(150°) , 150 and -210 are coterminal
= - cot(30°) , 150 is in II and by CAST rule cotangent is negative, and 30° from the x-axis
= -1/tan 30°
= -1/(1/√3) = -√3