Solve the equation on 0° ≤ θ < 360° and express in degrees to two decimal places.

sin(2θ) = -0.7843

I've gotten 308.345° (QIV) and 231.655° (QIII). I'm unsure how to get the answer though? The final answer is: 115.83°, 295.83°, 154.17°, 334.17°. Thank you to anyone who can help me!

well, you are correct, in that

2θ = 231.655° or 308.345°
That makes θ = 115.83° or 154.17°

However, since we're dealing with 2θ, that gives us answers in the 0<θ<180 range. To get all the values up to 360, we need to add 360 to 2θ (or, add 180 to θ) to our answers to get all values for 0<2θ<720, giving us the other two answers.

treat the 2Ø as one angle , let's say 2Ø = A

then sin A = -.7843
so we know that A is in III or IV , like you had

I then find the "angle in standard position" by finding
sin^-1 (+.7843) which is 51.67°

so in III, A = 180 + 51.66 = 231.66°
in IV , A = 360 - 51.66 = 308.34°
But A = 2Ø
2Ø = 231.66 -----> Ø = 115.83°
2Ø = 308.34 -----> Ø = 154.17°

Now we know that the period of sin 2Ø = 360/2 = 180
So by adding or subtracting 180 to any existing answer will yield a new answer
115.83 + 180 = 295.83°
154.17 + 180 = 334.17°

And that is how they got those 4 answers.

To solve the equation sin(2θ) = -0.7843, we need to find the values of θ that satisfy the equation within the given domain of 0° ≤ θ < 360°.

Step 1: Find the reference angle
Since sin is negative, we need to consider the solutions in QIII and QIV. Let's find the reference angle (θ') by taking the inverse sine of the absolute value of the given number:

θ' = sin^(-1)(0.7843)
θ' ≈ 51.78°

Step 2: Find the solutions in QIII and QIV
In QIII, the angle is 180° + reference angle:
θ_III = 180° + 51.78°
θ_III ≈ 231.78° (rounded to two decimal places)

In QIV, the angle is 360° - reference angle:
θ_IV = 360° - 51.78°
θ_IV ≈ 308.22° (rounded to two decimal places)

Step 3: Find all possible solutions within the given domain
To find all the possible solutions, we need to add multiples of 180° to the original solutions:

θ_1 = θ_III + k × 180°, where k is any integer
θ_2 = θ_IV + k × 180°, where k is any integer

Let's substitute k = 0, ±1, ±2, ... to find the solutions within the given domain:

For θ_1:
θ_1 = 231.78° + 0 × 180° ≈ 231.78°
θ_1 = 231.78° + 1 × 180° ≈ 411.78° (which is greater than 360°)

For θ_2:
θ_2 = 308.22° + 0 × 180° ≈ 308.22°
θ_2 = 308.22° + 1 × 180° ≈ 488.22° (which is greater than 360°)

Therefore, the solutions within the given domain are approximately:
θ = 231.78°, 308.22° (rounded to two decimal places)

Note: The other two values the user mentioned (115.83°, 295.83°, 154.17°, 334.17°) are not solutions to the given equation sin(2θ) = -0.7843.

To solve the equation sin(2θ) = -0.7843, we can use the inverse sine function.

Step 1: Take the inverse sine of both sides:
sin^(-1)(sin(2θ)) = sin^(-1)(-0.7843)

Step 2: Apply the double-angle identity for sine:
2θ = sin^(-1)(-0.7843)

Step 3: Divide both sides by 2:
θ = 0.5 * sin^(-1)(-0.7843)

Now, let's find the value of θ using a calculator:

Step 4: Calculate sin^(-1)(-0.7843) on your calculator.
The result is approximately -50.06°.

Step 5: Calculate θ:
θ = 0.5 * (-50.06°) = -25.03°

To express the answer in the desired range of 0° ≤ θ < 360°, we can add or subtract multiples of 360° to the obtained value.

Step 6: Add 360° to -25.03°:
-25.03° + 360° = 334.97°

So, one solution is θ = 334.97°.

Now, we can find the other solutions by using the symmetry of the sine function.

For any angle θ, sin(θ) = sin(180° - θ).

Step 7: Find the second solution:
180° - 334.97° = 154.17°

So, the second solution is θ = 154.17°.

To find the remaining solutions, we can use the periodicity of the sine function.

Step 8: Add 180° to each solution obtained so far:
334.97° + 180° ≈ 514.97°
154.17° + 180° = 334.17°

Therefore, the complete set of solutions on 0° ≤ θ < 360° is:
θ = 334.97°, 154.17°, 514.97°, 334.17°

Expressing each angle to two decimal places, we have:
θ ≈ 334.97° ≈ 335.00° (rounded to two decimal places)
θ ≈ 154.17° ≈ 154.17°
θ ≈ 514.97° ≈ 515.00°
θ ≈ 334.17° ≈ 334.17°

So, the final answers are approximately:
θ ≈ 335.00°, 154.17°, 515.00°, 334.17°