For each angle θ listed below, find the reference angle α, and then find sin θ. Round sin θ to four decimal places, if necessary.
θ = -110° ? ?
α = 180-110 = 70° in QIII
sin-110° = -sin70° = -0.93969
The reference angle α for θ = -110° is 70° in QIII. The sine of -110° is -0.9397.
To find the reference angle α for θ = -110°, follow these steps:
1. Determine whether θ lies in the first or fourth quadrant. Since θ is negative, it lies in the fourth quadrant.
2. Find the angle in the first quadrant that is coterminal with θ by adding 360° to θ:
θ + 360° = -110° + 360° = 250°
3. Subtract the result from 180° to find the reference angle:
α = 180° - 250° = -70°
Now let's find sin θ:
Since the reference angle α is negative, we can use the fact that the sine function is negative in the fourth quadrant.
Using a calculator or reference table, we can evaluate sin α to four decimal places:
sin α = sin(-70°) ≈ -0.9397
Therefore, sin θ ≈ -0.9397.
To find the reference angle α for θ = -110°, we need to determine an equivalent positive angle that has the same sine value as θ.
1. Identify the quadrant:
Since θ = -110°, it lies in the third quadrant.
2. Determine the reference angle:
For angles in the third quadrant, the reference angle can be found by subtracting the given angle from 180°. In this case:
α = 180° - |θ| = 180° - |-110°| = 180° - 110° = 70°
3. Calculate sin θ:
Since the sine function is periodic, we can find sin θ by finding sin α (the sine value of the reference angle) and taking into account the quadrant.
In the third quadrant, sine is negative, so we have:
sin θ = -sin α = -sin 70°
Now, using a scientific calculator or trigonometric table, find the sine of 70°. Rounding to four decimal places, we get:
sin 70° ≈ -0.9397
Therefore, sin θ ≈ -0.9397.