Find the values of x between (or equal to) 0 and 360 degrees to satisfy each equation.

arcsin1/2=x
I got 30 degrees as one angle, but how do I find the second one?

arccot1=x
Is the answer 45 degrees?

Also,

Find each value:

arccos(-1/2)

and

sin(2 sin^-1 1/2)

Thanks in advance. Even a little help is appreciated.

ALWAYS draw the problem

sketch your origin and x,y axes
now you can see that for a unit hypotenuse, the opposite side (y) is positive in guadrants one and two,\.
so 30 degrees is one answer
and 30 degrees above the -y axis is the other place where sin = 1/2
that is of course 180 - 30 = 150 degrees

now if cotangent is one, tangent is one.
so sure, 45 degrees will do.
BUT where else is y/x positive?
It is where both x and y are negative, quad 3
so 180 + 45 = 2556 is second answer

I think you can take it from there. Sketch your four quadrants and then your x and y

sines are positive in the first and second quadrant. If sin A = 1/2, A can be wither 30 or 150 degrees. The absolute value of sin, cost and tan etc. are determined by the angle with the horizontal axis.

cot and tan are negative in the second and fourth quadrants, and positive in the first and third quadrants.
cot^-1(1) = 45 or 225 degrees

cosine is negative in second and third quadrants.
cos-1(-1/2) = 120 and 240 degrees (60 degrees from the -x axis)

sin(2 sin^-1 1/2) = sin 2*30 or sin 2*150 = +0.866 or -0.866, since there are twqo possible values for sin^-1 (1/2)

"arcsin1/2=x

I got 30 degrees as one angle, but how do I find the second one?"

There is only one angle and it is 30 degrees. Don't confuse arcsin(1/2) with the set of the solutions of the equation sin(x) = 1/2.

It is similar to solving the equation:

x^2 = 4

and the squareroot function, in this case sqrt(4). The solution of the equation x^4 = 4 is not unique, there are two solutions: x = 2 and x = -2. So, there are two possible inverse functions that one can define. One has to make some choice. The squareroot function is defined as the postive solution of the equation. So sqrt(4) = 2 and not -2.

You can imagine what a terrible mess it would be if the two possible definitions were both used.

Similarly, one has defined the arcsin function such that it gives ONE of the solutions of the equation
sin(x) = y. By definition arcsin(y) is that solution of sin(x) = y that lies in the range between -pi/2 and pi/2.

but the question asked:

" Find the values of x between (or equal to) 0 and 360 degrees to satisfy each equation.

arcsin1/2=x "

Which is not exactly what is arcsin(1/2)

To find the second angle that satisfies the equation arcsin(1/2) = x, you need to consider the properties of the arcsin function. The arcsin function represents the inverse of the sine function, which means it will give you an angle whose sine is equal to the given value.

In this case, arcsin(1/2) = x means that the sine of angle x is equal to 1/2. Looking at the unit circle or the values on the unit circle, you can find the first angle that satisfies this equation, which is 30 degrees.

To find the second angle, you need to consider the periodic nature of the sine function. The sine function has a period of 360 degrees (or 2π radians). This means that after going through a full cycle, the sine values repeat.

In this case, since arcsin(1/2) = 30 degrees is a positive value, you know that the second angle will also have a positive sine value of 1/2. To find the angle, you can subtract the first angle from one full cycle of 360 degrees: 360 - 30 = 330 degrees.

So, the two angles that satisfy the equation arcsin(1/2) = x are 30 degrees and 330 degrees.

Moving on to the second equation, arccot(1) = x. The answer is not 45 degrees. The arccot function represents the inverse of the cotangent function, which means it will give you an angle whose cotangent is equal to the given value.

In this case, arccot(1) = x means that the cotangent of angle x is equal to 1. Looking at the unit circle or the values on the unit circle, you can find the angle whose cotangent is 1, which is 45 degrees.

So, the value that satisfies the equation arccot(1) = x is 45 degrees.

Moving on to the last two questions:

1. To find the value of arccos(-1/2), you need to find the angle whose cosine is equal to -1/2. Looking at the unit circle or the values on the unit circle, you can find that angle to be 120 degrees. Therefore, the value of arccos(-1/2) is 120 degrees.

2. To find the value of sin(2 sin^-1(1/2)), you need to evaluate the sine function at twice the value of the arcsine of 1/2. Since the arcsine of 1/2 is 30 degrees, you can substitute this value into the equation. So, sin(2(30 degrees)) = sin(60 degrees). The sine of 60 degrees is √3/2. Therefore, the value of sin(2 sin^-1(1/2)) is √3/2.

I hope this helps!