Math Problem Statement

(csc^2(-pi/4))/(-8sin(-pi/4))

Solution

Let's simplify the given expression:

csc2(π/4)8sin(π/4)\frac{\csc^2(-\pi/4)}{-8\sin(-\pi/4)}

Step-by-Step Solution:

  1. Evaluate csc2(π/4)\csc^2(-\pi/4):

    • Recall that csc(x)=1sin(x)\csc(x) = \frac{1}{\sin(x)}.
    • sin(π/4)=sin(π/4)=22\sin(-\pi/4) = -\sin(\pi/4) = -\frac{\sqrt{2}}{2}.
    • Therefore, csc(π/4)=1sin(π/4)=122=22\csc(-\pi/4) = \frac{1}{\sin(-\pi/4)} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}}.
    • Simplify 22-\frac{2}{\sqrt{2}} to 2-\sqrt{2}.
    • So, csc2(π/4)=(2)2=2\csc^2(-\pi/4) = (-\sqrt{2})^2 = 2.
  2. Evaluate sin(π/4)\sin(-\pi/4):

    • As noted, sin(π/4)=22\sin(-\pi/4) = -\frac{\sqrt{2}}{2}.
  3. Simplify the denominator 8sin(π/4)-8\sin(-\pi/4): 8sin(π/4)=8(22)=822=42-8\sin(-\pi/4) = -8 \cdot \left(-\frac{\sqrt{2}}{2}\right) = 8 \cdot \frac{\sqrt{2}}{2} = 4\sqrt{2}

  4. Put it all together: csc2(π/4)8sin(π/4)=242\frac{\csc^2(-\pi/4)}{-8\sin(-\pi/4)} = \frac{2}{4\sqrt{2}}

  5. Simplify the fraction: 242=122\frac{2}{4\sqrt{2}} = \frac{1}{2\sqrt{2}}

  6. Rationalize the denominator: 122=12222=24\frac{1}{2\sqrt{2}} = \frac{1 \cdot \sqrt{2}}{2\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{4}

Final Answer:

24\boxed{\frac{\sqrt{2}}{4}}

Would you like any further details or have questions on any steps?

Related Questions:

  1. How would you simplify csc2(π/4)\csc^2(\pi/4) without using negative angles?
  2. Can you explain why sin(x)=sin(x)\sin(-x) = -\sin(x)?
  3. How do you rationalize a denominator, and why is it necessary?
  4. What is the general process for simplifying trigonometric expressions?
  5. How do the even-odd identities affect trigonometric functions?

Tip:

When working with trigonometric functions of negative angles, remember that sin(x)=sin(x)\sin(-x) = -\sin(x) and csc(x)=csc(x)\csc(-x) = -\csc(x) because sine is an odd function.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Trigonometric Identities
Simplification of Expressions

Formulas

csc(x) = 1/sin(x)
sin(-x) = -sin(x)
Rationalizing the denominator

Theorems

Odd-Even Function Theorem
Simplification of Trigonometric Functions

Suitable Grade Level

Grades 10-12