Question

Without using a calculator, simplify the following expression to one trigonometry ratio: sin140°×sin(360°- x)/Cos50°×tan(-x)

Answer 1

To simplify the expression \( \sin 140^\circ \times \sin(360^\circ - x) / (\cos 50^\circ \times \tan(-x)) \), we can start by using trigonometric identities.

1. **Simplify \( \sin(360^\circ - x) \)**:
\[
\sin(360^\circ - x) = -\sin(x)
\]

2. **Simplify \( \tan(-x) \)**:
\[
\tan(-x) = -\tan(x)
\]

3. **Substituting these identities into the expression**:
\[
\sin 140^\circ \times \sin(360^\circ - x) = \sin 140^\circ \times (-\sin x) = -\sin 140^\circ \sin x
\]
\[
\tan(-x) = -\tan(x) \quad \text{so the denominator becomes} \quad \cos 50^\circ \times (-\tan x) = -\cos 50^\circ \tan x
\]

4. **Combine the terms**:
\[
\frac{-\sin 140^\circ \sin x}{-\cos 50^\circ \tan x} = \frac{\sin 140^\circ \sin x}{\cos 50^\circ \tan x}
\]

5. **Recall \( \tan x = \frac{\sin x}{\cos x} \)**:
Substituting this into the expression gives:
\[
\tan x = \frac{\sin x}{\cos x} \Rightarrow \frac{\sin 140^\circ \sin x}{\cos 50^\circ \cdot \frac{\sin x}{\cos x}} = \frac{\sin 140^\circ \sin x \cdot \cos x}{\cos 50^\circ \sin x}
\]

6. **Simplify the expression**:
Since \( \sin x \) in the numerator and denominator cancels out (assuming \( \sin x \neq 0 \)):
\[
\frac{\sin 140^\circ \cos x}{\cos 50^\circ}
\]

7. **Recognize that \( \sin 140^\circ \)** can be expressed in terms of its corresponding angle:
\[
\sin 140^\circ = \sin(180^\circ - 40^\circ) = \sin 40^\circ
\]
Thus:
\[
\frac{\sin 40^\circ \cos x}{\cos 50^\circ}
\]

8. **Use the identity \( \cos x = \sin(90^\circ - x) \)**, and knowing that \( \cos 50^\circ = \sin 40^\circ \), we substitute:
\[
\frac{\sin 40^\circ \cos x}{\sin 40^\circ} = \cos x
\]

Therefore, the simplified expression can be expressed as:
\[
\boxed{\cos x}
\]