Half Number Identities - 4 Examples

TL;DR

Learn how to find cosine and sine of various angles using half-angle identities.

Transcript

in this problem we have to find the exact value of the cosine of pi over 12 using half number identities so the half number identity for cosine is cosine of x over 2 is equal to plus or minus the square root of 1 plus cosine x over two the one for sine is the same except you have a minus here instead of a plus so the first thing we have to do is fi... Read More

Key Insights

• 😒 Understanding the quadrant location and unit circle values is crucial in determining whether to use plus or minus in half-angle trigonometric identity formulas.
• 🔺 Manipulating the angle condition to find the corresponding x value is necessary for applying half-angle identities accurately.
• 👨‍💼 The Pythagorean trigonometric identity can be used to find missing trigonometric values in half-angle identity problems for cosine and sine.

Questions & Answers

Q: How do you determine whether to use plus or minus in the half-angle identity formulas?

You can analyze the quadrant location of the angle on the unit circle to decide if cosine or sine should be positive or negative, indicating whether to use plus or minus in the formulas.

Q: Why is it helpful to think of x over 2 as one-half x in half-angle identity problems?

Thinking of x over 2 as one-half x aids in determining the appropriate x value that fits the angle condition, simplifying the application of half-angle identities for cosine and sine calculations.

Q: How can the square root in the final answers be simplified further after using half-angle identities?

Multiplying by the conjugate or rationalizing the denominator can help simplify square roots in the final answers obtained from solving cosine and sine of half angles problems using trigonometric identities.

Summary & Key Takeaways

• Half-angle identities for cosine and sine are explained for angles like pi/12 and negative pi/8.

• Use quadrant information and unit circle values to determine whether to use plus or minus in the formulas.

• Examples of finding exact values of cosine and sine using the half-angle identities are demonstrated.