Evaluating the Cosine of 5π/3 Using the Unit Circle and Periodicity Property of Cosine Function

Cos(5π/3)

To evaluate the cosine of an angle, we can use the unit circle or the cosine function’s periodicity

To evaluate the cosine of an angle, we can use the unit circle or the cosine function’s periodicity.

In this case, we have cos(5π/3).

To use the unit circle, we start by drawing a circle with a radius of 1 unit and marking the point (1, 0) as the initial position on the x-axis.

Next, we need to find the angle 5π/3 on the unit circle. To do this, start from the positive x-axis (initial position) and rotate counterclockwise by an angle of 5π/3.

Divide 2π (one full revolution) into 3 parts. Each part will be 2π/3. Since we need to go 5 parts, we have 5(2π/3) = 10π/3.

So, we rotate 10π/3 counterclockwise from the initial position. This brings us to a point on the unit circle with coordinates (-1/2, -√3/2).

Finally, we can now find the cosine of 5π/3 using the x-coordinate of the point on the unit circle we found. In this case, the x-coordinate is -1/2. Therefore, cos(5π/3) = -1/2.

Alternatively, we can use the periodicity property of the cosine function. Since cos(x) has a period of 2π, we can subtract or add any multiple of 2π to an angle without changing its cosine value.

In this case, 5π/3 can be written as 3π + 2π/3, which is equivalent to an angle of 2π/3 on the unit circle.

For 2π/3, if you draw the unit circle and find the point on the circle that corresponds to an angle of 2π/3, you will also find that the x-coordinate is -1/2. Hence, we get the same result of cos(5π/3) = -1/2.

Both methods give us the same answer of cos(5π/3) = -1/2.

More Answers: