The Secrets of Evaluating cos(60) Using the Unit Circle and Trigonometric Identity

cos60

To evaluate cos(60), we can use the unit circle or the trigonometric identity

To evaluate cos(60), we can use the unit circle or the trigonometric identity.

Using the unit circle:
1. Draw a circle and mark the positive x-axis, which represents 0 degrees or 360 degrees.
2. Divide the circle into four quadrants.
3. Start at the positive x-axis and move anticlockwise to the point where the angle measures 60 degrees.
4. The x-coordinate of this point is equal to cos(60).

By locating the 60-degree angle on the unit circle, you will find that it intersects at the (1/2, √3/2) point on the circle. Since the x-coordinate represents cos(60), the answer is 1/2.

Using the trigonometric identity:
cos(60) can also be evaluated using the relationship between right triangles and the trigonometric functions.
1. Draw a right triangle with one angle measuring 60 degrees.
2. To find the cosine of 60, we need to determine the ratio of the adjacent side to the hypotenuse.
3. Label the sides of the triangle as follows:
– The side adjacent to the angle (60 degrees) is the base, denoted as b.
– The hypotenuse is labeled as h.
4. By applying the Pythagorean theorem, you can determine the values of b and h. The theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, we have:
h^2 = b^2 + (h/2)^2
Simplifying the equation, we get:
h^2 = 4b^2 + b^2
h^2 = 5b^2
h = b√5
5. Now we have the values of b and h, we can determine cos(60) by dividing the adjacent side (b) by the hypotenuse (h):
cos(60) = b / h = b / (b√5) = 1 / √5 = √5 / 5

Therefore, cos(60) is equal to both 1/2 and √5/5, where the latter is a simplified form.

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