4.3. If $$ 4 an heta+3=0 $$ and $$ heta \in\left[90^{\circ} ; 270^{\circ} ight] $$, determine with the aid of a diagram, and without the use of a calculator, the value of : 4.3.1. $$ \cos heta $$ 4.3.2. $$ 1-\sin ^{2} heta $$

Question

4.3. If $$ 4 	an 	heta+3=0 $$ and $$ 	heta \in\left[90^{\circ} ; 270^{\circ}ight] $$, determine with the aid of a diagram, and without the use of a calculator, the value of : 4.3.1. $$ \cos 	heta $$ 4.3.2. $$ 1-\sin ^{2} 	heta $$

UpStudy AI · Accepted Answer

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

$$
4 \tan \theta + 3 = 0
$$

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

Rearranging the equation gives:

$$
4 \tan \theta = -3
$$

Dividing both sides by 4:

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

### Step 2: Determine the Quadrant

Since $$\theta$$ is in the interval $$[90^{\circ}, 270^{\circ}]$$, we know that $$\theta$$ is in the second or third quadrant. In these quadrants, the tangent function is negative, which is consistent with our solution.

### Step 3: Use the Pythagorean Identity

We can use the identity:

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

Let $$\sin \theta = -3k$$ and $$\cos \theta = 4k$$ for some positive $$k$$. This is because in the second quadrant, $$\sin \theta$$ is positive and $$\cos \theta$$ is negative, while in the third quadrant, both are negative. However, since we are looking for $$\cos \theta$$ and $$\sin \theta$$ in the second quadrant, we will take $$\sin \theta = 3k$$ and $$\cos \theta = -4k$$.

### Step 4: Find $$k$$ using the Pythagorean Identity

Using the identity:

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

Substituting our expressions for $$\sin \theta$$ and $$\cos \theta$$:

$$
(3k)^2 + (-4k)^2 = 1
$$

Calculating this gives:

$$
9k^2 + 16k^2 = 1
$$

$$
25k^2 = 1
$$

$$
k^2 = \frac{1}{25}
$$

$$
k = \frac{1}{5}
$$

### Step 5: Calculate $$\sin \theta$$ and $$\cos \theta$$

Now substituting back to find $$\sin \theta$$ and $$\cos \theta$$:

$$
\sin \theta = 3k = 3 \cdot \frac{1}{5} = \frac{3}{5}
$$

$$
\cos \theta = -4k = -4 \cdot \frac{1}{5} = -\frac{4}{5}
$$

### Step 6: Answer the Questions

#### 4.3.1. Find $$\cos \theta$$

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

#### 4.3.2. Find $$1 - \sin^2 \theta$$

Using the identity $$1 - \sin^2 \theta = \cos^2 \theta$$:

$$
1 - \sin^2 \theta = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25}
$$

### Final Answers

- **4.3.1.** $$\cos \theta = -\frac{4}{5}$$
- **4.3.2.** $$1 - \sin^2 \theta = \frac{16}{25}$$
$$ \cos \theta = -\frac{4}{5}, \quad 1 - \sin^2 \theta = \frac{16}{25} $$

4.3. If and , determine with the aid of a diagram,
and without the use of a calculator, the value of :
4.3.1.
4.3.2.

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4.3. If and , determine with the aid of a diagram, and without the use of a calculator, the value of : 4.3.1. 4.3.2.