how do I find theta for the equation sin θ = -0.9063 with the given interval of (270 degrees, 360 degrees)
Set your calculator to DEG (degrees) if it isn't already set to it.
consider sin θ = + 0.9063
θ = 64.999, lets call it 65°
This is the angle in "standard position".
However, you are told that θ is in quadrant IV
so θ = 360-65 degrees
θ = 295°
verify by taking sin 295° on your calculator
To find theta in the equation sin θ = -0.9063 within the given interval of (270 degrees, 360 degrees), you can follow these steps:
Step 1: Start by finding the reference angle for the given value of -0.9063. Since the sine function is negative, the angle is in either the third or fourth quadrant. To find the reference angle, we can use the inverse sine function or arcsin:
arcsin(0.9063) ≈ 64.6 degrees
Step 2: Determine the angle with the reference angle. Since the sine is negative, the angle will be in the third quadrant. Therefore, the sum of the reference angle and 180 degrees will give us the angle in the third quadrant:
64.6 degrees + 180 degrees ≈ 244.6 degrees
Step 3: Check if the angle is within the given interval of (270 degrees, 360 degrees). In this case, 244.6 degrees is less than 270 degrees, so it is not within the interval.
Hence, there is no solution for the equation sin θ = -0.9063 within the interval (270 degrees, 360 degrees).
To find the value of theta (θ) for the equation sin(θ) = -0.9063 within the given interval of (270 degrees, 360 degrees), you can use the inverse sine function or arcsine.
1. Start by expressing -0.9063 as a decimal.
-0.9063
2. Use the inverse sine function (sin^(-1) or arcsin) to find the angle whose sine is -0.9063.
arcsin(-0.9063) ≈ -65.46 degrees
Note: The inverse sine function returns the angle in radians, but for simplicity, we will convert it to degrees in the next step.
3. Convert the angle from radians to degrees.
-65.46 degrees × (180 degrees / π radians) ≈ -65.46 degrees
4. Check if the obtained angle is within the given interval of (270 degrees, 360 degrees).
Since -65.46 degrees is smaller than the lower bound of 270 degrees, it is not within the specified interval.
Therefore, within the given interval, there is no value of theta (θ) that satisfies the equation sin(θ) = -0.9063.