Step 1: Understanding the Concept:
We are given the value of a trigonometric ratio and the quadrant in which the angle lies. We need to determine the values of other ratios (\(\cos\) and \(\tan\)) while respecting the sign conventions of the 4th quadrant.
: Key Formula or Approach:
1. Pythagoras Theorem: \( \text{Hypotenuse}^2 = \text{Base}^2 + \text{Perpendicular}^2 \).
2. Sign conventions: In the 4th quadrant, \(\cos \theta\) is positive, and \(\sin \theta\), \(\tan \theta\) are negative.
Step 2: Detailed Explanation:
Given \( \sin \theta = -24/25 \).
Let Perpendicular \( (P) = 24 \) and Hypotenuse \( (H) = 25 \).
Calculate Base \( (B) \):
\[ B = \sqrt{H^2 - P^2} = \sqrt{25^2 - 24^2} = \sqrt{625 - 576} = \sqrt{49} = 7 \]
Since \( \theta \) is in the 4th quadrant:
\[ \cos \theta = \frac{B}{H} = \frac{7}{25} \quad (\text{Positive in 4th quadrant}) \]
\[ \tan \theta = \frac{P}{B} = -\frac{24}{7} \quad (\text{Negative in 4th quadrant}) \]
Now, substitute these into the expression:
\[ E = 7 \tan \theta + 25 \cos \theta \]
\[ E = 7 \left( -\frac{24}{7} \right) + 25 \left( \frac{7}{25} \right) \]
\[ E = -24 + 7 = -17 \].
Step 3: Final Answer:
The value of the expression is -17.