Find all angles θ between 0° and 180° satisfying the given equation. (Enter your answers as a comma-separated list. Round your answers to one decimal place.)

Sin θ = 1/4

You know that the sine is positive in quads I and II, which is your given domain

if sinØ = 1/4
Ø = 14.5° or Ø = 180 - 14.5 = 165.5°

Find all the angles 𝜃 between 0° and 180° satisfying sin 𝜃 =

√3/2

Well, let's break out the ol' Sin θ = 1/4. First, let me tell you a secret. Sin θ is like a really picky eater, it only likes to hang out with a select few numbers. Lucky for us, one of those numbers is 1/4.

Now, to find the angles θ that make Sin θ = 1/4, we have to go on a little math adventure. Buckle up!

Since Sin θ = 1/4, we're looking for angles where the opposite side of a right triangle is 1 and the hypotenuse is 4. I'm always up for a right triangle road trip!

Using our trusty Pythagorean Theorem, we can find the adjacent side. Let's call it x.

4^2 = x^2 + 1^2

Squaring things is always fun. So, 16 = x^2 + 1.

Now, let's solve for x. Subtract 1 from both sides: x^2 = 15.

Taking the square root of both sides (careful, it's contagious), we get x = √15.

Now, remember, we're looking for angles θ between 0° and 180°.

To find those angles, we have to take the inverse Sin (or Sin^-1) of 1/4.

Using a fancy calculator or computer, we find that Sin^-1(1/4) ≈ 14.5°.

But wait, there's more! Since Sin is a repeat offender, it loves to hang out with other angles that make it happy.

So, we can also find the second solution by subtracting 14.5° from 180° (a.k.a. the cool kid's club).

180° - 14.5° ≈ 165.5°.

And there you have it! The angles θ that make Sin θ = 1/4 are approximately 14.5° and 165.5°.

So, grab some popcorn and enjoy the show, because Sin θ has found its perfect match!

To find all angles θ between 0° and 180° satisfying the equation sin θ = 1/4, we can use the inverse sine function (also known as arcsine or sin^(-1)).

1. Start by taking the inverse sine of both sides of the equation: sin^(-1)(sin θ) = sin^(-1)(1/4).
This simplifies to: θ = sin^(-1)(1/4).

2. Use a calculator to find the inverse sine of 1/4. Most scientific calculators will have a button labeled "sin^(-1)" or "arcsin" that you can use to find the inverse sine. If you're using a computer calculator or an online calculator, look for the function labeled "asin" or "arcsin".

3. Enter 1/4 into the calculator or online calculator and find its inverse sine. The result will be an angle in radians.

4. Convert the angle from radians to degrees by multiplying it by 180/π (since there are 180° in π radians).

5. Round the result to one decimal place and write it down. This is one of the angles θ that satisfies the equation sin θ = 1/4.

6. Repeat steps 3-5 if there are any other values for θ within the given range (0° to 180°).

By following these steps, you can find all angles θ between 0° and 180° that satisfy the equation sin θ = 1/4.