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 = 255° ? ?
To find the reference angle α for 255°, we need to find the acute angle that 255° makes with the x-axis.
Since 255° is in the third quadrant, the reference angle can be found by subtracting 180° from 255°.
α = 255° - 180° = 75°
After finding the reference angle, we can then determine sin θ.
Since sin is positive in the second and third quadrants, and 255° is in the third quadrant, we need to consider the negative value of sin.
sin 255° = -sin α
Now, we need to find the sine of 75°.
Using a calculator, sin 75° ≈ 0.9659.
Therefore, sin 255° ≈ -0.9659 (rounded to four decimal places).
To find the reference angle α for an angle θ, follow these steps:
1. Determine which quadrant the angle θ falls in.
- In this case, θ = 255°, which falls in the third quadrant (180° < θ < 270°).
2. Subtract the angle from 360° to find the reference angle.
- α = 360° - 255° = 105°
Now let's find sin θ by using the reference angle α:
3. Determine the sign of sin θ based on the quadrant.
- In the third quadrant, sin θ is negative.
4. Calculate sin α using the reference angle.
- Since sin α = sin 105°, we need to use a calculator or trigonometric table to determine sin 105°.
Rounding the result to four decimal places, sin θ ≈ -0.9700