Evaluate the expression.
arcsin (-1/2)
Is there a rule I am missing? How do I know which quadrant the triangle belongs in? That is the only thing that is stopping me. Much help appreciated.
where is the sine negative?
Isn't it in quadrants III and IV ?
The sine of which angle is 1/2 ?
Isn't it 30º or pi/6 radians
so arcsin (-1/2) is 210º or 330º
which would be 7pi/6 or 11pi/6 radians
Sorry, my mistake. It is arccos, not arcsin. Also, the negative in the book is exactly in the middle. The book just says evaluate the expression.
first of all
-1/2 = 1/-2 = -(1/2)
it makes no difference where the negative sign is
just repeat my steps from above, except use the properties of Cosine
1. the cosine is negative in the 2nd and 3rd quadrants.
2. cos (60 degrees) = cos pi/3 = 1/2
(hint: one of the answers is 2pi/3)
But how do I know where cosine (or tangent and sine) are positive? I don't understand.
Thanks for the help, btw.
It is called the CAST rule
this page explains it well
(Broken Link Removed)
It all makes sense now!!!! Thanks, haha.
To evaluate the expression arcsin(-1/2), we can use the concept of trigonometric functions and the unit circle.
The arcsin function, or inverse sine function, returns the angle whose sine is equal to a given value. In this case, we want to find the angle whose sine is -1/2.
To determine the quadrant the triangle belongs in, we need to consider the sign of both the x and y coordinates in the unit circle.
In the coordinate plane, the unit circle is centered at the origin (0,0) with a radius of 1 unit. All points on the unit circle have the form (cosθ, sinθ), where θ represents the angle measured from the positive x-axis.
For arcsin(-1/2), the y-coordinate is -1/2. Since the sine function is negative in the third and fourth quadrants, the angle will be located in one of those quadrants.
To determine which quadrant exactly, we can refer to the x-coordinate. In this case, since arcsin(-1/2) is negative, we know that the x-coordinate will be negative.
Combining these two pieces of information, we conclude that the angle whose sine is -1/2 is in the third quadrant, where both the x and y coordinates are negative.
To find the actual value of arcsin(-1/2), we can use the unit circle or refer to a trigonometric table. In the third quadrant, the sine value is negative, so we need to find the angle whose sine has an absolute value of 1/2.
In the unit circle, we observe that the angle with a sine value of 1/2 in the third quadrant is π/6 radians or 30 degrees.
Therefore, arcsin(-1/2) is equal to -π/6 radians or -30 degrees.
So, the evaluation of the expression arcsin(-1/2) is -π/6 radians or -30 degrees.