How to Find the Exact Value of tan(pi/12)

TL;DR

To find the exact value of tan(pi/12), use the difference formula for tangent by expressing pi/12 as pi/3 minus pi/4. This results in tan(pi/3) - tan(pi/4) divided by 1 plus tan(pi/3) times tan(pi/4), ultimately simplifying to 2 - sqrt(3) or -sqrt(3) + 2.

Transcript

hi in this problem we're going to find the exact value of the tangent of pi over 12. so to do this we're going to start by writing pi over 12 as either a sum or difference of two more familiar angles 4 minus 3 is 1 and so that makes me think that we can write pi over 12 as 4 pi over 12 minus 3 pi over 12. and these have been carefully chosen becaus... Read More

Key Insights

• 🍹 Writing pi/12 as a sum or difference of two familiar angles helps in applying trigonometric formulas.
• 👻 The difference formula for the tangent function allows for the calculation of the tangent of the difference of two angles.
• 😑 Rationalizing the denominator is a common technique to simplify trigonometric expressions.
• ❓ The exact value of the tangent of pi/12 is -sqrt(3) + 2 or 2 - sqrt(3).

Questions & Answers

Q: How can pi/12 be written as a sum or difference of two familiar angles?

Pi/12 can be written as 4pi/12 minus 3pi/12, where 4pi/12 is equal to pi/3 and 3pi/12 is equal to pi/4.

Q: What is the difference formula for the tangent function?

The difference formula for the tangent function states that tangent of (x-y) is equal to tangent of x minus tangent of y, divided by 1 plus tangent x tangent y.

Q: How can the tangent of pi/12 be simplified?

The tangent of pi/12 can be simplified by rationalizing the denominator. By multiplying by 1 minus the square root of 3 over 1 minus the square root of 3, the denominator can be simplified to -2, resulting in the value -sqrt(3) + 2.

Q: Can the answer be left in the form sqrt(3) - 1?

Yes, leaving the answer as sqrt(3) - 1 is acceptable and considered to be simplified. However, some prefer to write it as 2 - sqrt(3).

Summary & Key Takeaways

• The video demonstrates how to write pi/12 as a difference of two familiar angles, pi/3 and pi/4.

• It explains the difference formula for the tangent function and how to apply it to the problem.

• The video shows step-by-step calculations to find the exact value of the tangent of pi/12.