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 = 405° ? ?
To find the reference angle α, we need to determine the acute angle formed between the terminal side of θ and the x-axis.
Since 405° is more than 360°, we need to subtract 360° from it until we obtain an angle between 0° and 360°.
405° - 360° = 45°
Therefore, the reference angle α = 45°.
To find sin θ, we will look at the corresponding reference angle α.
The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.
Since we know that the reference angle α = 45°, we can construct a right triangle and label the sides. Let's name the opposite side as "opposite" and the hypotenuse as "hypotenuse".
By using the Pythagorean theorem, we can determine the length of the adjacent side as follows:
adjacent^2 + opposite^2 = hypotenuse^2
adjacent^2 + 45^2 = 1^2
adjacent^2 = 1^2 - 45^2
adjacent^2 = 1 - 2025
adjacent^2 = -2024
Since we obtained a negative value for the adjacent side, this means that the angle is not within the usual 0° to 360° range in the coordinate plane. In this case, we can determine the value of sine using a reference angle in the fourth quadrant.
The sine in the fourth quadrant is the same as the sine in the second quadrant (but negative). We can use the sine of the reference angle in the second quadrant to find the sine of the reference angle in the fourth quadrant.
sin α = sin 45° = 0.7071
Therefore, sin θ = -0.7071 (rounded to four decimal places).
To find the reference angle α, we need to determine the acute angle formed between the given angle θ and the x-axis.
Step 1: Identify the related acute angle
Since the given angle θ = 405° lies in the fourth quadrant (between 360° and 450°), we need to subtract it from 360° to find the related acute angle.
Related acute angle α = 360° - θ
= 360° - 405°
= -45°
However, since this angle is negative, we add 360° to make it positive.
Related acute angle α = -45° + 360°
= 315°
Step 2: Find the sine of the angle
The sine of an angle is the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle.
In this case, since we have an angle in the fourth quadrant, the reference angle α lies in the first quadrant where sine is positive.
sin θ = sin α = sin 315°
≈ 0.7071 (rounded to four decimal places)
Therefore, sin θ ≈ 0.7071.