How do you figure out trigonometric functions that aren't on the unit circle?
example: sin(-27pi/6)
sin(-27pi/6)
= sin(-(27/6)π ±2kπ)
Adjust the value of k so that the expression within the parentheses is within 0 to 2π.
In this particular case, putting k=+3 will give
sin(-27pi/6) = sin((3/2)π)
When k is an integer
sin(x) = (sin x + 2k*pi)
set k=2 and...
sin(-27pi/6) = sin(-27pi/6 + 2*2*pi)
= sin(-27pi/6 + 24pi/6)
= sin(-pi/2)
= -1
To find the value of trigonometric functions for angles that are not on the unit circle, you can use the periodicity and symmetry properties of trigonometric functions.
For example, in the given example of sin(-27π/6), we can simplify it first:
-27π/6 = -9π/2
Now, let's look at the properties involved:
1. Periodicity: The sine function has a period of 2π, which means that sin(x + 2π) = sin(x). We can utilize this property when dealing with large or small angles that are outside the domain of the unit circle.
2. Symmetry: The sine function is an odd function, which means sin(-x) = -sin(x). This symmetry can be helpful when working with negative angles.
Using these properties, we can simplify the given angle, -9π/2, to an equivalent angle within the unit circle:
-9π/2 = -4π - (π/2)
Since -4π is an integral multiple of 2π (2π * -2), we can ignore it in terms of determining the value of the trigonometric function.
Therefore, we are left with -π/2, which is the same as -90 degrees. Now we can use the values of the trigonometric functions for common angles:
sin(-90°) = -sin(90°) = -1
So, sin(-27π/6) equals -1.