Exact Value of sin(pi/12)

TL;DR

Calculating sine of π/12 by applying difference of angles and trigonometric identities.

Transcript

hello in this problem we are going to find  the exact value of the sine of pi over 12.   so to do this we're going to write pi over 12   as a difference of two angles that we're more  familiar with in particular if you have pi over 12 you can write it as 4 pi over 12 minus 3 pi over 12. and this is really convenient because  we actually know what t... Read More

Key Insights

• 🆘 Utilizing known angles helps simplify trigonometric calculations.
• 😑 Trigonometric identities aid in manipulating complex expressions for easier solving.
• ❓ Memorizing trigonometric values is crucial for accurate and efficient calculations.

Questions & Answers

Q: How is π/12 expressed as π/3 - π/4 for the sine calculation?

π/12 is written as π/3 - π/4, leveraging the familiarity of these angles through dividing π by 3 and 4 respectively.

Q: Which trigonometric identity is used to determine the sine of π/12?

The difference identity for sine, which states sin(x - y) = sin(x)cos(y) - cos(x)sin(y), is applied to calculate the sine of π/12.

Q: Why is it crucial to accurately substitute the values into the trigonometric functions?

Accurate substitution of π/3 and π/4 values into sine and cosine functions ensures the correct evaluation of sine(π/12) without errors in the solution.

Q: How is the final value of sine(π/12) expressed after simplifying the calculation?

The sine of π/12 is simplifed to √6 - √2 / 4, attained by multiplying and combining fractions with the same denominator.

Summary & Key Takeaways

• Convert π/12 into π/3 - π/4 using known angles.

• Apply sine difference identity to simplify calculation.

• Evaluate sine(π/3) and cosine(π/4) to find the exact value of sine(π/12).