Verify the Trigonometric Identity cos(3pi/2 - x) = -sin(x)

TL;DR

Cosine of (3pi/2 - X) equals negative sine of X through trigonometric identity proof.

Transcript

hello in this video we're going to show that the cosine of 3 pi over 2 minus X is equal to minus sine X let's go ahead and carefully work through it solution we'll start by writing down the formula that we're going to use for this problem that tells us if we have the cosine of a minus B this is equal to the cosine of a times the cosine of B and the... Read More

Key Insights

• ❓ Trigonometric identity: cos(3pi/2 - X) = -sin(X).
• 🔙 Utilizes cos(A - B) formula: cos(A)cos(B) + sin(A)sin(B).
• 🦻 Unit circle aids in determining cosine and sine values for 3pi/2.
• 🥺 Simplification leads to conclusion of -sin(X).
• 🈸 Application extends to other angles for deriving similar identities.
• ❓ Understanding trigonometric functions enhances problem-solving.
• 👨‍💼 Memorization technique for cosine and sine computation.

Questions & Answers

Q: How is the trigonometric identity for cos(3pi/2 - X) = -sin(X) derived?

The identity follows from the cos(A - B) formula, which yields cos(A)cos(B) + sin(A)sin(B). By substituting A as 3pi/2 and B as X, the equation simplifies to -sin(X).

Q: What role does the unit circle play in determining cosine and sine values for 3pi/2?

The unit circle assists in locating the angle 3pi/2, corresponding to the point (0, -1). This guides the evaluation of cos(3pi/2) as 0 and sin(3pi/2) as -1.

Q: How does the proof of cos(3pi/2 - X) = -sin(X) conclude?

By substituting cos(3pi/2) and sin(3pi/2) into the formula and simplifying, the result shows that the expression equals -sin(X), fulfilling the trigonometric identity.

Q: Can the trigonometric identity be extended to other angles besides 3pi/2?

Yes, the same logic can be applied to cos(A - B) with different values of A and B to derive similar trigonometric identities, showcasing the versatility of trigonometric formulas.

Summary & Key Takeaways

• Cosine of (3pi/2 - X) is proven to be equal to negative sine of X using trigonometric identities.

• Utilizes the formula for cos(A - B) to derive the identity.

• Demonstrates application of unit circle values for cosine and sine calculations.