Need on these 2 questions and my last 2 uploads !

19.

20. The work is the answer

Step-by-step explanation:

19. We begin with the equation (for my ease, I will swap theta with x)

First, we can use the Pythagorean Identity to substitute in for

This gives us

Next, we can subtract 1 from each side

Now, we can divide each side by

Now we move the negative to the other side

Now we can do the inverse of sin to find our value

With this inverse, we are looking for an angle that has the sin (or the y value) of -1. As this is an inverse sin function, it has the domain restriction of

.

From this information, we find that

But this answer does not fit the initial interval that we were given. Luckily, is the same location as , so that is our answer.

20. We begin with the expression and we have to prove that it is equal to

First, we need to use the Difference of 2 Angles Trig Identity. This says:

When we apply this to our expression, we get

Next, we can simplify our the two finite aspects

When we plug in these two values, we get

This simplifies to which is what we were trying to verify.

the range is 19

step-by-step explanation:

the range is biggest number minus the smallest, so in this case it would be

41 - 22 = 19

51-50

step-by-step explanation: