Question

Solve without a calculator

Sin 140 times sin360minus x all over cos50 times tan negative x

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Answer 1

To solve the expression \(\frac{\sin 140^\circ \cdot \sin(360^\circ - x)}{\cos 50^\circ \cdot \tan(-x)}\), we will simplify each part step-by-step.

### Step 1: Simplify \(\sin(360^\circ - x)\)

Using the sine subtraction identity:
\[
\sin(360^\circ - x) = -\sin x
\]

### Step 2: Substitute into the expression

Now replace \(\sin(360^\circ - x)\) in the original expression:
\[
\frac{\sin 140^\circ \cdot (-\sin x)}{\cos 50^\circ \cdot \tan(-x)}
\]

### Step 3: Simplify \(\tan(-x)\)

Using the property of tangent:
\[
\tan(-x) = -\tan(x)
\]

### Step 4: Substitute into the expression

Now, replace \(\tan(-x)\) in the expression:
\[
\frac{\sin 140^\circ \cdot (-\sin x)}{\cos 50^\circ \cdot (-\tan x)} = \frac{\sin 140^\circ \cdot \sin x}{\cos 50^\circ \cdot \tan x}
\]

### Step 5: Simplify \(\tan x\) and \(\cos 50^\circ\)

We know:
\[
\tan x = \frac{\sin x}{\cos x}
\]
Substituting this, we have:
\[
\frac{\sin 140^\circ \cdot \sin x}{\cos 50^\circ \cdot \frac{\sin x}{\cos x}} = \frac{\sin 140^\circ \cdot \sin x \cdot \cos x}{\cos 50^\circ \cdot \sin x}
\]

### Step 6: Cancel \(\sin x\)

Provided \(\sin x \neq 0\), we can cancel \(\sin x\):
\[
\frac{\sin 140^\circ \cdot \cos x}{\cos 50^\circ}
\]

### Step 7: Simplify \(\sin 140^\circ\)

We know:
\[
\sin 140^\circ = \sin(180^\circ - 40^\circ) = \sin 40^\circ
\]

### Step 8: Substitute \(\sin 140^\circ\)

Now substitute \(\sin 140^\circ\) in the expression:
\[
\frac{\sin 40^\circ \cdot \cos x}{\cos 50^\circ}
\]

### Step 9: Use \(\cos 50^\circ\) identity

Using the co-function identity:
\[
\cos 50^\circ = \sin 40^\circ
\]
Thus, the expression simplifies further to:
\[
\frac{\sin 40^\circ \cdot \cos x}{\sin 40^\circ}
\]

### Step 10: Final Simplification

Again, provided \(\sin 40^\circ \neq 0\), we can cancel \(\sin 40^\circ\):
\[
\cos x
\]

### Conclusion

Thus, the final result of the given expression is:
\[
\cos x
\]