how sin 5π/3 =to √3/2
To show that sin(5π/3) is equal to √3/2, we can start by determining the reference angle for the angle 5π/3.
In the unit circle, 5π/3 is in the third quadrant. The reference angle is the angle formed between the positive x-axis and the terminal side of the angle within the first quadrant. To find the reference angle, we subtract 5π/3 from 2π (full revolution).
2π - 5π/3 = 6π/3 - 5π/3 = π/3
The reference angle for 5π/3 is π/3.
Since the reference angle is π/3 and sin(π/3) = √3/2, we can conclude that sin(5π/3) = √3/2.
To understand how sin(5π/3) equals √3/2, let's break it down step by step.
Step 1: Convert 5π/3 to degrees.
To convert from radians to degrees, we can use the formula: degrees = (radians * 180) / π.
Let's calculate the value of 5π/3 in degrees:
5π/3 * 180 / π = 300°
So, 5π/3 equals 300°.
Step 2: Identify the reference angle.
In trigonometry, the reference angle is the acute angle between the terminal side of an angle and the x-axis in standard position.
To find the reference angle for 300°, we subtract it from the nearest multiple of 360°:
360° - 300° = 60°
Thus, the reference angle for 300° is 60°.
Step 3: Determine the value of sin(60°).
In a standard unit circle, sin(60°) is equal to √3/2. This can be memorized or derived from the ratios of the sides of a 30-60-90 right triangle, where sin(60°) = opposite/hypotenuse = √3/2.
So, sin(60°) equals √3/2.
Step 4: Apply the periodicity of sine function.
Since the sine function has a periodicity of 360° (or 2π radians), we can infer that sin(360° + 60°) will have the same value as sin(60°).
Step 5: Apply the knowledge to sin(5π/3).
Since 5π/3 is equivalent to 300°, and sin(60°) is √3/2, we can conclude that sin(5π/3) is also equal to √3/2.
Therefore, sin(5π/3) = √3/2.