Determine whether each equation is true for all x for which both sides of the equation are defined. If it is true, support your conclusion with a sketch using the unit circle. If it is false, give a counterexample.

inverse sin(-x)=-inverse sin(x)
inverse cos(-x)=-inverse cos(x)

I think that the first one is true and the second one is false but I do not know what to put down as a sketch or counterexample. Any help is appreciated.

The simplest way to determine if

inverse f(-x) =-inverse f(x)
is true for all values of x is to find out if f(x)=f(-x) for all values of x.

Substitute sin(x) or cos(x) for f(x) above and do a little sketching of each of the two trigonometric functions between -pi and +pi. You will get your answer by simply looking at the sketches.

To determine whether each equation is true or false, we need to understand the properties of inverse trigonometric functions and evaluate them using the unit circle.

1. inverse sin(-x) = -inverse sin(x):

First, let's consider the range of the inverse sine function. The inverse sine function, also known as arcsine, takes an output between -π/2 and π/2. This means that the principal value of inverse sin(x) lies between -π/2 and π/2.

Now, let's consider the behavior of the inverse sine function. When we negate the input, the output will also be negated. In other words, if sin(x) = y, then sin(-x) = -y.

Therefore, inverse sin(-x) will have the same value as -inverse sin(x) for all x where both sides of the equation are defined. This means that the equation is true for all x.

To support this conclusion with a sketch using the unit circle, plot the points (x, inverse sin(-x)) and (-x, -inverse sin(x)) on the unit circle. Notice that they have the same vertical coordinate (y-value). This demonstrates that the equation holds true.

2. inverse cos(-x) = -inverse cos(x):

Similar to the first equation, let's consider the range of the inverse cosine function. The inverse cosine function, also known as arccosine, takes an output between 0 and π. This means that the principal value of inverse cos(x) lies between 0 and π.

Now, consider the behavior of inverse cosine under negation. When we negate the input, the output remains the same. In other words, if cos(x) = y, then cos(-x) = y.

Therefore, inverse cos(-x) will have the same value as -inverse cos(x) for all x where both sides of the equation are defined.

To provide a counterexample for this equation, we need to find an x such that inverse cos(-x) ≠ -inverse cos(x). Let's take x = π/6 as an example.

inverse cos(-π/6) = inverse cos(π/6) = π/6
However, -inverse cos(π/6) = -π/6

Since inverse cos(-π/6) ≠ -inverse cos(π/6), we have found a counterexample for this equation.

To visualize this counterexample, plot the point (π/6, inverse cos(-π/6)) on the unit circle and (-π/6, -inverse cos(π/6)). Notice that they have different horizontal coordinates (x-values), indicating that the equation is false.

In conclusion, the first equation inverse sin(-x) = -inverse sin(x) is true for all values of x, while the second equation inverse cos(-x) = -inverse cos(x) is false, as demonstrated by the counterexample.