Math

QuestionEvaluate cotπ3\cot \frac{\pi}{3} using special right triangles. Simplify your answer with integers or fractions.

Studdy Solution

STEP 1

Assumptions1. The function cot\cot is the cotangent function, which is the reciprocal of the tangent function. . The angle π3\frac{\pi}{3} is in radians, not degrees.
3. We are using special right triangles, specifically the30-60-90 triangle, to evaluate the expression.

STEP 2

First, we need to understand that the cotangent of an angle in a right triangle is the ratio of the adjacent side to the opposite side.cotθ=adjacentopposite\cot \theta = \frac{adjacent}{opposite}

STEP 3

In a30-60-90 triangle, the length of the side opposite the60-degree angle is 3\sqrt{3} times the length of the side opposite the30-degree angle, and the length of the hypotenuse is twice the length of the side opposite the30-degree angle.

STEP 4

We can use these relationships to evaluate the cotangent of π3\frac{\pi}{3}, which is the same as the cotangent of60 degrees.

STEP 5

The cotangent of60 degrees is the ratio of the adjacent side (opposite the30-degree angle) to the opposite side (opposite the60-degree angle).
cot60=adjacentopposite\cot60^{\circ} = \frac{adjacent}{opposite}

STEP 6

In a30-60-90 triangle, the length of the side opposite the30-degree angle is1 (if we normalize the triangle), and the length of the side opposite the60-degree angle is 3\sqrt{3}.

STEP 7

Substitute these values into the equation.
cot60=13\cot60^{\circ} = \frac{1}{\sqrt{3}}

STEP 8

Therefore, the cotangent of π3\frac{\pi}{3} (or60 degrees) is 13\frac{1}{\sqrt{3}}.
cotπ3=13\cot \frac{\pi}{3} = \frac{1}{\sqrt{3}}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord