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To find the values of [tex]\(\sin \Theta\)[/tex] and [tex]\(\tan \Theta\)[/tex] given that [tex]\(\cos \Theta = -\frac{4}{7}\)[/tex]:
### Step 1: Find [tex]\(\sin \Theta\)[/tex]
We use the Pythagorean identity, which states:
[tex]\[ \sin^2 \Theta + \cos^2 \Theta = 1 \][/tex]
Given [tex]\(\cos \Theta = -\frac{4}{7}\)[/tex], we substitute this into the identity:
[tex]\[ \sin^2 \Theta + \left(-\frac{4}{7}\right)^2 = 1 \][/tex]
[tex]\[ \sin^2 \Theta + \frac{16}{49} = 1 \][/tex]
To isolate [tex]\(\sin^2 \Theta\)[/tex], subtract [tex]\(\frac{16}{49}\)[/tex] from both sides:
[tex]\[ \sin^2 \Theta = 1 - \frac{16}{49} \][/tex]
[tex]\[ \sin^2 \Theta = \frac{49}{49} - \frac{16}{49} \][/tex]
[tex]\[ \sin^2 \Theta = \frac{33}{49} \][/tex]
Now, take the square root of both sides to find [tex]\(\sin \Theta\)[/tex]:
[tex]\[ \sin \Theta = \pm \sqrt{\frac{33}{49}} \][/tex]
[tex]\[ \sin \Theta = \pm \frac{\sqrt{33}}{7} \][/tex]
Since [tex]\(\sqrt{33} \approx 5.74\)[/tex], this simplifies to approximately:
[tex]\[ \sin \Theta \approx \pm 0.821 \][/tex]
Thus, the possible values for [tex]\(\sin \Theta\)[/tex] are:
[tex]\[ \sin \Theta = 0.8206518066482898 \quad \text{(positive value)} \][/tex]
[tex]\[ \sin \Theta = -0.8206518066482898 \quad \text{(negative value)} \][/tex]
### Step 2: Find [tex]\(\tan \Theta\)[/tex]
The tangent function is defined as:
[tex]\[ \tan \Theta = \frac{\sin \Theta}{\cos \Theta} \][/tex]
Using the positive value for [tex]\(\sin \Theta = 0.8206518066482898\)[/tex]:
[tex]\[ \tan \Theta = \frac{0.8206518066482898}{-\frac{4}{7}} \][/tex]
[tex]\[ \tan \Theta = 0.8206518066482898 \times \left(-\frac{7}{4}\right) \][/tex]
[tex]\[ \tan \Theta = -1.4361406616345074 \][/tex]
Similarly, using the negative value for [tex]\(\sin \Theta = -0.8206518066482898\)[/tex]:
[tex]\[ \tan \Theta = \frac{-0.8206518066482898}{-\frac{4}{7}} \][/tex]
[tex]\[ \tan \Theta = -0.8206518066482898 \times \left(-\frac{7}{4}\right) \][/tex]
[tex]\[ \tan \Theta = 1.4361406616345074 \][/tex]
### Summary
Given [tex]\(\cos \Theta = -\frac{4}{7}\)[/tex], the possible values for [tex]\(\sin \Theta\)[/tex] and [tex]\(\tan \Theta\)[/tex] are:
[tex]\[ \sin \Theta \approx 0.8206518066482898 \quad \text{or} \quad -0.8206518066482898 \][/tex]
[tex]\[ \tan \Theta \approx -1.4361406616345074 \quad \text{or} \quad 1.4361406616345074 \][/tex]
### Step 1: Find [tex]\(\sin \Theta\)[/tex]
We use the Pythagorean identity, which states:
[tex]\[ \sin^2 \Theta + \cos^2 \Theta = 1 \][/tex]
Given [tex]\(\cos \Theta = -\frac{4}{7}\)[/tex], we substitute this into the identity:
[tex]\[ \sin^2 \Theta + \left(-\frac{4}{7}\right)^2 = 1 \][/tex]
[tex]\[ \sin^2 \Theta + \frac{16}{49} = 1 \][/tex]
To isolate [tex]\(\sin^2 \Theta\)[/tex], subtract [tex]\(\frac{16}{49}\)[/tex] from both sides:
[tex]\[ \sin^2 \Theta = 1 - \frac{16}{49} \][/tex]
[tex]\[ \sin^2 \Theta = \frac{49}{49} - \frac{16}{49} \][/tex]
[tex]\[ \sin^2 \Theta = \frac{33}{49} \][/tex]
Now, take the square root of both sides to find [tex]\(\sin \Theta\)[/tex]:
[tex]\[ \sin \Theta = \pm \sqrt{\frac{33}{49}} \][/tex]
[tex]\[ \sin \Theta = \pm \frac{\sqrt{33}}{7} \][/tex]
Since [tex]\(\sqrt{33} \approx 5.74\)[/tex], this simplifies to approximately:
[tex]\[ \sin \Theta \approx \pm 0.821 \][/tex]
Thus, the possible values for [tex]\(\sin \Theta\)[/tex] are:
[tex]\[ \sin \Theta = 0.8206518066482898 \quad \text{(positive value)} \][/tex]
[tex]\[ \sin \Theta = -0.8206518066482898 \quad \text{(negative value)} \][/tex]
### Step 2: Find [tex]\(\tan \Theta\)[/tex]
The tangent function is defined as:
[tex]\[ \tan \Theta = \frac{\sin \Theta}{\cos \Theta} \][/tex]
Using the positive value for [tex]\(\sin \Theta = 0.8206518066482898\)[/tex]:
[tex]\[ \tan \Theta = \frac{0.8206518066482898}{-\frac{4}{7}} \][/tex]
[tex]\[ \tan \Theta = 0.8206518066482898 \times \left(-\frac{7}{4}\right) \][/tex]
[tex]\[ \tan \Theta = -1.4361406616345074 \][/tex]
Similarly, using the negative value for [tex]\(\sin \Theta = -0.8206518066482898\)[/tex]:
[tex]\[ \tan \Theta = \frac{-0.8206518066482898}{-\frac{4}{7}} \][/tex]
[tex]\[ \tan \Theta = -0.8206518066482898 \times \left(-\frac{7}{4}\right) \][/tex]
[tex]\[ \tan \Theta = 1.4361406616345074 \][/tex]
### Summary
Given [tex]\(\cos \Theta = -\frac{4}{7}\)[/tex], the possible values for [tex]\(\sin \Theta\)[/tex] and [tex]\(\tan \Theta\)[/tex] are:
[tex]\[ \sin \Theta \approx 0.8206518066482898 \quad \text{or} \quad -0.8206518066482898 \][/tex]
[tex]\[ \tan \Theta \approx -1.4361406616345074 \quad \text{or} \quad 1.4361406616345074 \][/tex]
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