Question
Find the exact value (a) \( \cot 20^{\circ}-\frac{\sin 70^{\circ}}{\sin 20^{\circ}} \)
Ask by Hanson Murphy. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The exact value of the expression \( \cot 20^{\circ} - \frac{\sin 70^{\circ}}{\sin 20^{\circ}} \) is **0**.
Solution
To find the exact value of the expression:
\[
\cot 20^{\circ} - \frac{\sin 70^{\circ}}{\sin 20^{\circ}}
\]
Let's break it down step by step.
1. **Express \(\cot 20^{\circ}\) in terms of sine and cosine:**
\[
\cot 20^{\circ} = \frac{\cos 20^{\circ}}{\sin 20^{\circ}}
\]
2. **Recognize that \(\sin 70^{\circ}\) is equivalent to \(\cos 20^{\circ}\):**
\[
\sin 70^{\circ} = \sin (90^{\circ} - 20^{\circ}) = \cos 20^{\circ}
\]
3. **Substitute \(\sin 70^{\circ}\) with \(\cos 20^{\circ}\) in the original expression:**
\[
\cot 20^{\circ} - \frac{\sin 70^{\circ}}{\sin 20^{\circ}} = \frac{\cos 20^{\circ}}{\sin 20^{\circ}} - \frac{\cos 20^{\circ}}{\sin 20^{\circ}} = 0
\]
Thus, the exact value of the expression is:
**Answer:** 0
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Simplify this solution Mind Expander
To find the exact value of \( \cot 20^{\circ}-\frac{\sin 70^{\circ}}{\sin 20^{\circ}} \), we can simplify the expression step by step: 1. Recall that \( \sin 70^{\circ} = \cos 20^{\circ} \) since \( \sin(90^{\circ} - x) = \cos x \). 2. Thus, we can rewrite the expression: \[ \cot 20^{\circ} - \frac{\cos 20^{\circ}}{\sin 20^{\circ}}. \] 3. The cotangent function can be expressed as: \[ \cot 20^{\circ} = \frac{\cos 20^{\circ}}{\sin 20^{\circ}}. \] 4. This means our expression simplifies to: \[ \frac{\cos 20^{\circ}}{\sin 20^{\circ}} - \frac{\cos 20^{\circ}}{\sin 20^{\circ}} = 0. \] Thus, the exact value is: \[ \boxed{0}. \]
