Exact Value of cos(x/2) given sin(x) = -4/5 and x in (3pi/2, 2pi) using Half-Angle Identities

TL;DR

Given sine X = -4/5 and X between 3π/2 and 2π, find cos(X/2) using trig identities.

Transcript

find the exact value of cosine of X over two given that sine X is equal to negative four-fifths and that X is between 3 PI over 2 and 2 pi solution so in this problem we have to use the half number identities so the half number identities for cosine are cosine of X over 2 equals plus or minus the square root of 1 plus cosine X all over to the 1/4 s... Read More

Key Insights

• 🔺 Utilize half-angle identities for trigonometric functions when dealing with angles less than 180 degrees.
• 🤘 The quadrant of the angle on the unit circle assists in determining the sign in trigonometric calculations.
• 🦻 Pythagorean identity aids in finding the accurate values of trigonometric functions.
• 🔺 Manipulating angles to half angles simplifies trigonometric calculations.

Questions & Answers

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

To decide between plus or minus, analyze the quadrant where X/2 lies on the unit circle. Negative x-coordinate implies a negative cosine, therefore, choose the negative square root.

Q: How do you find the exact value of cosine X/2?

Firstly, calculate cosine X by using the Pythagorean identity with sine X. Then, apply the half-angle identity for cosine to obtain the precise value of cosine X/2.

Q: Why is it necessary to manipulate the given angle X to X/2?

Manipulating X to X/2 allows for the application of the half-angle identities for trigonometric functions, simplifying the calculation process for determining cosine X/2 accurately.

Q: Why is it crucial to analyze the position of X on the unit circle?

Understanding the position of X on the unit circle is vital as it determines the sign (positive or negative) to be used in the half-angle trigonometric calculations, ensuring the correct evaluation of cosine X/2.

Summary & Key Takeaways

• Given sine X = -4/5 and X between 3π/2 and 2π.

• Use half-angle identities for cosine.

• Determine whether to use plus or minus for cosine X/2 and calculate the exact value.