EXERCISE 3 If $$ 5 \sin heta+4=0 $$ and $$ \cos heta0 $$, find the value of $$ an heta \cdot \cos heta $$.

Question

EXERCISE 3 If $$ 5 \sin 	heta+4=0 $$ and $$ \cos 	heta>0 $$, find the value of $$ 	an 	heta \cdot \cos 	heta $$.

UpStudy AI · Accepted Answer

To solve the problem, we start with the equation given:

$$
5 \sin \theta + 4 = 0
$$

### Step 1: Solve for $$\sin \theta$$

Rearranging the equation, we get:

$$
5 \sin \theta = -4
$$

Dividing both sides by 5:

$$
\sin \theta = -\frac{4}{5}
$$

### Step 2: Determine $$\cos \theta$$

Since we know that $$\sin^2 \theta + \cos^2 \theta = 1$$, we can find $$\cos \theta$$:

$$
\cos^2 \theta = 1 - \sin^2 \theta
$$

Calculating $$\sin^2 \theta$$:

$$
\sin^2 \theta = \left(-\frac{4}{5}\right)^2 = \frac{16}{25}
$$

Now substituting this into the equation for $$\cos^2 \theta$$:

$$
\cos^2 \theta = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25}
$$

Taking the square root to find $$\cos \theta$$:

$$
\cos \theta = \pm \sqrt{\frac{9}{25}} = \pm \frac{3}{5}
$$

### Step 3: Determine the correct sign for $$\cos \theta$$

We are given the condition that $$\cos \theta > 0$$. Therefore, we take the positive value:

$$
\cos \theta = \frac{3}{5}
$$

### Step 4: Find $$\tan \theta$$

The tangent function is defined as:

$$
\tan \theta = \frac{\sin \theta}{\cos \theta}
$$

Substituting the values we found:

$$
\tan \theta = \frac{-\frac{4}{5}}{\frac{3}{5}} = -\frac{4}{3}
$$

### Step 5: Calculate $$\tan \theta \cdot \cos \theta$$

Now we can find $$\tan \theta \cdot \cos \theta$$:

$$
\tan \theta \cdot \cos \theta = \left(-\frac{4}{3}\right) \cdot \left(\frac{3}{5}\right)
$$

Calculating this gives:

$$
\tan \theta \cdot \cos \theta = -\frac{4 \cdot 3}{3 \cdot 5} = -\frac{4}{5}
$$

### Final Answer

Thus, the value of $$\tan \theta \cdot \cos \theta$$ is:

$$
\boxed{-\frac{4}{5}}
$$
$$\tan \theta \cdot \cos \theta = -\frac{4}{5}$$

EXERCISE 3 If \( 5 \sin \theta+4=0 \) and \( \cos \theta>0 \), find the value of \( \tan \theta \cdot \cos \theta \).