6.2 Determine, without using a calculator, the value of the following trigonometric
expression:

Ask by Mcfarlane Rogers. in South Africa

Mar 19,2025

Question

6.2 Determine, without using a calculator, the value of the following trigonometric
expression:

Ask by Mcfarlane Rogers. in South Africa

Mar 19,2025

UpStudy AI · Accepted Answer

We start with the given expression:
$$
\frac{\sin 2x \cdot \cos (-x) + \cos 2x \cdot \sin\left(360^\circ - x\right)}{\sin\left(180^\circ + x\right)}
$$

**Step 1. Simplify the numerator:**

Use the properties of trigonometric functions:
- The cosine function is even, so:
  $$
  \cos(-x) = \cos x
  $$
- The sine function has the periodic property:
  $$
  \sin\left(360^\circ - x\right) = -\sin x
  $$
  
Substitute these into the numerator:
$$
\sin 2x \cdot \cos x + \cos 2x \cdot \left(-\sin x\right) = \sin 2x \cdot \cos x - \cos 2x \cdot \sin x
$$

**Step 2. Recognize the sine angle subtraction formula:**

Recall the sine difference formula:
$$
\sin(A - B) = \sin A \cos B - \cos A \sin B
$$
  
Identify $$A = 2x$$ and $$B = x$$. Hence:
$$
\sin 2x \cos x - \cos 2x \sin x = \sin(2x - x) = \sin x
$$
  
Thus, the numerator simplifies to:
$$
\sin x
$$

**Step 3. Simplify the denominator:**

Recall the sine function property for angle addition:
$$
\sin(180^\circ + x) = -\sin x
$$

So the denominator becomes:
$$
-\sin x
$$

**Step 4. Combine the numerator and the denominator:**

The expression now is:
$$
\frac{\sin x}{-\sin x}
$$

As long as $$\sin x \neq 0$$, we can cancel $$\sin x$$ in the numerator and denominator:
$$
\frac{\sin x}{-\sin x} = -1
$$

Thus, the value of the given trigonometric expression is:
$$
-1
$$
The value of the expression is $$-1$$.

6.2 Determine, without using a calculator, the value of the following trigonometric
expression:

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6.2 Determine, without using a calculator, the value of the following trigonometric expression:

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Solución

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6.2 Determine, without using a calculator, the value of the following trigonometric
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