To solve for x, I need to take the inverse sine of both sides of the equation. For values in which this function is defined, it will cancel out the sine. I make this assumption.

sin^-1(sin(x)) = sin^-1(0.25)

x = sin^-1(0.25)

Now, I take out my calculator and evaluate, making sure that my calculator is in radian mode. 

The first result I get, rounding to the nearest hundredth is:

x ≈ 0.25

Since 0.25 radians is a solution, I know that 0.25 + 2π represents all multiples of 0.25, and thus many more solutions to the equation. 

Next, I need to consider that there is another solution that isn’t a multiple of 0.25, which I can find with the formula:

π – sin^-1(0.25)

Approximated, this is 2.89 radians. 

All the solutions that hadn’t been accounted for by 0.25 + 2π are thus considered with the expression 2.89 + 2π.

I am given an interval over which to restrict my answer: between 0 and 2π. So, I need to make sure I list out all the valid answers that are between that range.

To make it easier, I estimate what 2π is.

2π ≈ 6.28

So anything between 0 and 6.28 radians is valid.

First, I’ll start by considering my initial answer of 0.25 radians. Since 0 < 0.25 < 6.28, it is a solution.

Next, I’ll consider if any multiple of 0.25 will also be a solution. It is obvious that it won’t, since 0.25 + 2π, or 0.25 + 6.28, is always greater than 2π, or 6.28. 

Now, I’ll decide whether my second determined answer, 2.89 radians, works. Since 0 < 2.89 < 6.28, it does. However, no multiple of 2.89 will work since 2.89 + 6.28 > 6.28.

The solutions to the equation sin(x) = 0.25 for the interval 0 ≤ x ≤ 2π, rounded to the nearest hundredth, are x = 0.25, 2.89 radians.

I can use my calculator (in radian mode) to evaluate sin(x) for x = 0.25 and x = 2.89 to determine if the result is equal to 0.25. 

sin(0.25) ≈ 0.25

sin(2.89) ≈ 0.25

I got the answers right.

I start by isolating x in the equation.

9cos(6x) – 4 = 5

9cos(6x) = 9

cos(6x) = 1

I remove the cos() by taking the cos^-1 (inverse cosine) of both sides. Here, I must assume the inverse cosine function is defined.

cos(6x) = 1

6x = cos^-1(1)

x = cos^-1(1)/6

I use my calculator to evaluate cos^-1(1)/6.

cos^-1(1)/6 = 0

Now, I need to write the solution.

Generally, the solutions to an equation involving cosine are the solutions determined with the statement cos(𝜃) = cos(-𝜃), + 360˚ • n, where n is any integer. However, since I’m finding the solutions to cos^-1(1)/6, I need to divide 360 by 6. This results in the solution statement:

0 + 60˚ • n

This is also simply 60˚ • n.

The solutions to the equation are 60˚ • n, where n is any integer.

I can plug in 60˚, 120˚, and 180˚ in for x in the original equation to check if I have solved the equation correctly.

Using my calculator (in degree mode), I get:

9cos(6(60˚)) – 4 = 5

9cos(6(120˚)) – 4 = 5

9cos(6(180˚)) – 4 = 5

I got the answer right.