$$ \frac { \sin 210 ^ { \circ } \cdot \sin 225 ^ { \circ } \cdot \cos ( heta - 180 ^ { \circ } ) } { \sin 135 ^ { \circ } \cdot \operatorname { cog } heta } $$

Question

$$ \frac { \sin 210 ^ { \circ } \cdot \sin 225 ^ { \circ } \cdot \cos ( 	heta - 180 ^ { \circ } ) } { \sin 135 ^ { \circ } \cdot \operatorname { cog } 	heta } $$

UpStudy AI · Accepted Answer

We start with the expression

$$
\frac{ \sin 210^\circ \cdot \sin 225^\circ \cdot \cos(\theta-180^\circ)}{\sin 135^\circ \cdot \cos \theta},
$$

assuming that “cog $$\theta$$” was meant to be $$\cos\theta$$.

**Step 1. Evaluate the constant trigonometric terms**

1. $$\sin 210^\circ$$:

$$
\sin210^\circ = \sin(180^\circ+30^\circ) = -\sin30^\circ = -\frac{1}{2}.
$$

2. $$\sin 225^\circ$$:

$$
\sin225^\circ = \sin(180^\circ+45^\circ) = -\sin45^\circ = -\frac{\sqrt{2}}{2}.
$$

3. $$\sin 135^\circ$$:

$$
\sin135^\circ = \sin(180^\circ-45^\circ) = \sin45^\circ = \frac{\sqrt{2}}{2}.
$$

**Step 2. Simplify $$\cos(\theta-180^\circ)$$**

Using the cosine difference formula,

$$
\cos(\theta-180^\circ) = \cos\theta\cos180^\circ + \sin\theta\sin180^\circ.
$$

Since

$$
\cos180^\circ = -1 \quad \text{and} \quad \sin180^\circ = 0,
$$

we obtain

$$
\cos(\theta-180^\circ) = \cos\theta(-1) + \sin\theta(0) = -\cos\theta.
$$

**Step 3. Substitute the values back into the expression**

Replace all trigonometric values:

$$
\text{Numerator} = \left(-\frac{1}{2}\right) \cdot \left(-\frac{\sqrt{2}}{2}\right) \cdot \left(-\cos\theta\right).
$$

Compute the product of the constants first:

$$
-\frac{1}{2} \cdot -\frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4}.
$$

Multiplying by $$-\cos\theta$$ gives

$$
\frac{\sqrt{2}}{4} \cdot (-\cos\theta) = -\frac{\sqrt{2}}{4}\cos\theta.
$$

Thus, the numerator is $$-\frac{\sqrt{2}}{4}\cos\theta$$ and the denominator is $$\frac{\sqrt{2}}{2}\cos\theta$$.

The expression becomes

$$
\frac{-\frac{\sqrt{2}}{4}\cos\theta}{\frac{\sqrt{2}}{2}\cos\theta}.
$$

**Step 4. Cancel the common factors**

Assuming $$\cos\theta \neq 0$$ (so we can cancel $$\cos\theta$$) and cancel $$\sqrt{2}$$:

$$
\frac{-\frac{\sqrt{2}}{4}}{\frac{\sqrt{2}}{2}} = -\frac{1}{4} \cdot \frac{2}{1} = -\frac{2}{4} = -\frac{1}{2}.
$$

**Final Answer**

$$
-\frac{1}{2}.
$$
The expression simplifies to $$-\frac{1}{2}$$.

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