I don't understand how sin(5pi/3)= -squareroot(3)/2 and not +squareroot(3)/2. 5pi/3 is equal 300degree. it is on the +x-axis so it exact value should be positive right?Explain.
5 pi /3 is 300 degrees indeed
that is 60 degrees BELOW the + x axis which is at 0 or 360
so x is positive and y is NEGATIVE
so the sin = y/x is negative
sin = y/r
y is negative
r is always positive
To understand why sin(5π/3) is equal to -√3/2 and not +√3/2, let's first review the unit circle and the trigonometric functions.
In a unit circle, the radius is 1, and angles are measured by the amount of rotation counter-clockwise from the positive x-axis. The coordinates of any point on the unit circle can be determined using the trigonometric functions sine (sin), cosine (cos), and tangent (tan).
Now, let's look at the angle 5π/3, which is equivalent to 300 degrees. To find the coordinates of the point on the unit circle corresponding to this angle, we can follow the steps below:
1. Determine the reference angle: The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. For 5π/3, the reference angle is π/3 (60 degrees).
2. Determine the quadrant: To determine the sign of the trigonometric function, we need to identify the quadrant in which the angle lies. For 5π/3, the angle falls in the fourth quadrant.
3. Determine the sign: In the fourth quadrant, only the sine function is positive, while the cosine and tangent functions are negative. Hence, sin(5π/3) will be positive, and cos(5π/3) and tan(5π/3) will be negative.
4. Calculate the values: The reference angle of π/3 (60 degrees) lies on the unit circle at the coordinates (√3/2, 1/2). Since the angle is in the fourth quadrant with negative cosine and tangent, the coordinates of the point for 5π/3 become (-√3/2, -1/2).
Therefore, sin(5π/3) is equal to -√3/2 since the sine function is positive in the fourth quadrant.
Remember that the signs of trigonometric functions can vary depending on the quadrant in which the angle lies. It's essential to follow these steps to determine the correct sign while working with angles on the unit circle.