Solve sin ¥è = -0.204 for 90¨¬ < ¥è < 270¨¬. Give your answer to the nearest tenth of a degree.
we want θ such that sin θ = -0.204
sine is negative in the 3rd & 4th quadrants, so we will only have one answer between 90° and 270°
Arcsin .204 = 11.77°
so, we want 180° + 11.77° = 191.77°
To solve the equation sin ¥è = -0.204 for 90¨¬ < ¥è < 270¨¬, you can follow these steps:
Step 1: Convert the given value of sin ¥è into an angle measure.
You can use the inverse sine function (sin^(-1)) to find the angle whose sine is -0.204.
sin^(-1)(-0.204) = -11.73¨¬
(Note: The inverse sine function gives the principal value, which lies between -90¨¬ and 90¨¬.)
Step 2: Determine the reference angle.
The reference angle can be found by taking the absolute value of the angle calculated above:
Reference angle = |-11.73¨¬| = 11.73¨¬
Step 3: Determine the possible angles within the given range.
Since the sin function is negative in the third and fourth quadrants, we need to find the angle in the third and fourth quadrants with the same reference angle.
For the third quadrant (180¨¬ < ¥è < 270¨¬):
¥è = 180¨¬ + Reference angle = 180¨¬ + 11.73¨¬ = 191.73¨¬
For the fourth quadrant (270¨¬ < ¥è < 360¨¬):
¥è = 360¨¬ - Reference angle = 360¨¬ - 11.73¨¬ = 348.27¨¬
Therefore, the two solutions within the given range are approximately 191.7¨¬ and 348.3¨¬.