Simplizy $$ \frac{\cos (90+ heta)}{\sin \left( heta-180^{\circ} ight)+3 \sin (- heta)} $$

Question

Simplizy $$ \frac{\cos (90+	heta)}{\sin \left(	heta-180^{\circ}ight)+3 \sin (-	heta)} $$

UpStudy AI · Accepted Answer

We start with the expression

$$
\frac{\cos (90^\circ+\theta)}{\sin (\theta-180^\circ)+3\sin (-\theta)}
$$

**Step 1. Simplify the Numerator**

Recall the trigonometric identity

$$
\cos (90^\circ+\theta) = -\sin \theta.
$$

Thus, the numerator becomes

$$
-\sin \theta.
$$

**Step 2. Simplify the Denominator**

We simplify each term in the denominator separately.

1. For the first term, use the identity for sine with a shifted angle:

$$
\sin(\theta-180^\circ) = -\sin \theta.
$$

2. For the second term, use the identity for the sine of a negative angle:

$$
\sin(-\theta) = -\sin \theta.
$$

Thus, multiplying by 3 gives:

$$
3\sin(-\theta) = -3\sin \theta.
$$

Now, the denominator simplifies to

$$
-\sin \theta - 3\sin \theta = -4\sin \theta.
$$

**Step 3. Combine the Results**

Substitute the simplified numerator and denominator back into the expression:

$$
\frac{-\sin \theta}{-4\sin \theta}.
$$

Assuming $$\sin \theta \neq 0$$ (to avoid division by zero), the $$-\sin \theta$$ cancels with $$-4\sin \theta$$ as follows:

$$
\frac{-\sin \theta}{-4\sin \theta} = \frac{1}{4}.
$$

**Final Answer**

$$
\frac{\cos (90^\circ+\theta)}{\sin (\theta-180^\circ)+3\sin (-\theta)} = \frac{1}{4}, \quad \text{for } \sin \theta \neq 0.
$$
$$ \frac{\cos (90^\circ+\theta)}{\sin (\theta-180^\circ)+3\sin (-\theta)} = \frac{1}{4}, \quad \text{for } \sin \theta \neq 0. $$

Simplizy \( \frac{\cos (90+\theta)}{\sin \left(\theta-180^{\circ}\right)+3 \sin (-\theta)} \)

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Trigonometry Questions