Math Problem Statement

sin

𝜋 4

cos

𝜋 3

− sin

𝜋 3

cos

𝜋 4

2

Solution

The expression provided can be solved as follows:

Given:

sin(π4)cos(π3)sin(π3)cos(π4)\sin\left(\frac{\pi}{4}\right)\cos\left(\frac{\pi}{3}\right) - \sin\left(\frac{\pi}{3}\right)\cos\left(\frac{\pi}{4}\right)

Step 1: Recognize it as a trigonometric identity

This matches the form of the sine difference identity: sin(AB)=sin(A)cos(B)cos(A)sin(B)\sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B)

Here: A=π4,B=π3A = \frac{\pi}{4}, \, B = \frac{\pi}{3}

So, the expression simplifies to: sin(π4π3)\sin\left(\frac{\pi}{4} - \frac{\pi}{3}\right)

Step 2: Simplify the argument

π4π3=3π4π12=π12\frac{\pi}{4} - \frac{\pi}{3} = \frac{3\pi - 4\pi}{12} = -\frac{\pi}{12}

Thus, the expression becomes: sin(π12)\sin\left(-\frac{\pi}{12}\right)

Step 3: Simplify the sine function

Using the property of sine: sin(x)=sin(x)\sin(-x) = -\sin(x)

So: sin(π12)=sin(π12)\sin\left(-\frac{\pi}{12}\right) = -\sin\left(\frac{\pi}{12}\right)

Step 4: Approximation (if needed)

The exact value of sin(π12)\sin\left(\frac{\pi}{12}\right) can be derived geometrically or approximated numerically. If approximating: sin(π12)0.2588\sin\left(\frac{\pi}{12}\right) \approx 0.2588

Thus: sin(π12)0.2588\sin\left(-\frac{\pi}{12}\right) \approx -0.2588

Would you like detailed steps on calculating sin(π12)\sin\left(\frac{\pi}{12}\right) exactly or using its expansion?


Further Exploration

  1. What are the derivations of the sine addition and subtraction formulas?
  2. How can sin(π12)\sin\left(\frac{\pi}{12}\right) be calculated using half-angle identities?
  3. How do sine and cosine behave for negative angles?
  4. What are alternative methods for evaluating trigonometric expressions?
  5. How can we generalize such expressions for any AA and BB?

Tip: Trigonometric identities often simplify complex-looking expressions—always check for a pattern or formula that applies!

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Sine and Cosine Functions
Trigonometric Identities

Formulas

sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
sin(-x) = -sin(x)

Theorems

Sine Difference Identity

Suitable Grade Level

Grades 9-12