How do you evaluate the expression tan(8) * f * cos(8) * 13 * 8 in Quadrant III and sin(f) * 14 * f in Quadrant II?

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To evaluate this expression involving angles in different quadrants, we first need to understand the trigonometric properties in each quadrant.

1. **Understanding the Quadrants**: In Quadrant III, both sine and cosine values are negative, while in Quadrant II, sine values are positive, and cosine values are negative.

2. **Evaluate Individual Components**:

  • tan(8): The tangent function is the ratio of sine to cosine. To compute tan(8 degrees, for instance), we calculate tan(8) = sin(8) / cos(8).
  • f in Quadrant III: Since f is in Quadrant III, both sin(f) and cos(f) are negative. Hence, tan(f) = sin(f) / cos(f) will be positive.
  • cos(8): The cosine of 8 degrees is positive, hence cos(8) ≈ 0.9903.
  • sin(f): In Quadrant II, sin(f) for an angle f will be positive. Thus, if we take sin(f) for f = 14 degrees ≈ 0.2419.

3. **Combining the Expressions**:

The expression can now be evaluated component-wise:

– For the first part: tan(8) * f * cos(8) * 13 * 8

– For the second part: sin(f) * 14 * f

4. **Putting it All Together**: After substituting the values computed above:

  • Calculate tan(8) * f * cos(8) * 13 * 8

    • using the values for tan(8) and cos(8).

  • Calculate sin(f) * 14 * f using the value of sin(f).

5. **Final Calculation**: Finally, sum or multiply as required by the expression to get the final result.

Remember that precision matters, especially when working with trigonometric values. Using a calculator will provide precise values for sin and cos which could enhance accuracy. Always check your results for consistency within the expected range of values!

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