Question

How does sin(2π/3) = cos(π/6)?

a) Using the angle sum identity for sine
b) Utilizing the unit circle
c) Applying the Pythagorean identity
d) Employing the reciprocal trigonometric functions

Answer 1

To show that sin(2π/3) = cos(π/6), we can use two approaches: the angle sum identity for sine and the unit circle.

Step-by-step explanation:

To show that sin(2π/3) = cos(π/6), we can use two approaches: the angle sum identity for sine and the unit circle.

a) Using the angle sum identity for sine:

1. sin(2π/3) = sin(π/3 + π/3)
2. Using the angle sum identity, sin(2π/3) = sin(π/3)cos(π/3) + cos(π/3)sin(π/3)
3. Since sin(π/3) = √3/2 and cos(π/3) = 1/2, we can substitute these values in the equation.
4. Therefore, sin(2π/3) = (√3/2)(1/2) + (1/2)(√3/2) = √3/4 + √3/4 = (2√3)/4 = √3/2.

b) Utilizing the unit circle:

1. We know that sin(2π/3) represents the y-coordinate of a point on the unit circle.
2. To find the y-coordinate, we need to determine the angle that corresponds to π/3. This angle is located in the second quadrant of the unit circle.
3. The cosine function represents the x-coordinate of a point on the unit circle.
4. In the second quadrant, the x-coordinate is negative. Therefore, cos(π/6) = -1/2.
5. So, sin(2π/3) = -1/2.

Therefore, sin(2π/3) = cos(π/6), either by using the angle sum identity for sine or by utilizing the unit circle.