Math

QuestionFind the exact value of cos(π3)\cos \left(-\frac{\pi}{3}\right) using reference angles.

Studdy Solution

STEP 1

Assumptions1. We are working with trigonometric functions, specifically the cosine function. . The angle given is in radians, not degrees.
3. We are using the concept of reference angles to find the exact value of the expression.
4. The reference angle for an angle in standard position is the acute angle formed by the terminal side of the angle and the x-axis.

STEP 2

First, we need to understand that the cosine function is positive in the first and fourth quadrants. Since the given angle is negative, we know it's in the fourth quadrant.

STEP 3

The reference angle for any angle is always positive. To find the reference angle for a negative angle, we take the absolute value of the given angle.
Referenceangle=π3Reference\, angle = |\, -\frac{\pi}{3}\, |

STEP 4

Calculate the reference angle.
Referenceangle=π3Reference\, angle = \frac{\pi}{3}

STEP 5

Now, we use the reference angle to find the exact value of the cosine function. Since the reference angle is in the fourth quadrant where cosine is positive, the cosine of the given angle is the same as the cosine of the reference angle.
cos(π3)=cos(π3)\cos \left(-\frac{\pi}{3}\right) = \cos \left(\frac{\pi}{3}\right)

STEP 6

The exact value of cosine of pi/3 is known from the unit circle or the special right triangles.cos(π3)=12\cos \left(\frac{\pi}{3}\right) = \frac{1}{2} So, the exact value of the given expression is1/2.

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