Use a Pythagorean identity to find [tex]\(\sin (\theta)\)[/tex] if [tex]\(\cos (\theta) = \frac{2}{7}\)[/tex] and the terminal side of [tex]\(\theta\)[/tex] lies in quadrant IV.

[tex]\[
\sin (\theta) = \square
\][/tex]



Answer :

To find [tex]\(\sin(\theta)\)[/tex] given that [tex]\(\cos(\theta) = \frac{2}{7}\)[/tex] and the terminal side of [tex]\(\theta\)[/tex] lies in quadrant IV, you can use the Pythagorean identity. The Pythagorean identity states:

[tex]\[ \sin^2(\theta) + \cos^2(\theta) = 1 \][/tex]

Given [tex]\(\cos(\theta) = \frac{2}{7}\)[/tex], plug it into the identity:

[tex]\[ \sin^2(\theta) + \left(\frac{2}{7}\right)^2 = 1 \][/tex]

Calculate [tex]\(\left(\frac{2}{7}\right)^2\)[/tex]:

[tex]\[ \left(\frac{2}{7}\right)^2 = \frac{4}{49} \][/tex]

Now, substitute it back into the identity:

[tex]\[ \sin^2(\theta) + \frac{4}{49} = 1 \][/tex]

To isolate [tex]\(\sin^2(\theta)\)[/tex], subtract [tex]\(\frac{4}{49}\)[/tex] from both sides:

[tex]\[ \sin^2(\theta) = 1 - \frac{4}{49} \][/tex]

Rewrite 1 as a fraction with the same denominator:

[tex]\[ 1 = \frac{49}{49} \][/tex]

So,

[tex]\[ \sin^2(\theta) = \frac{49}{49} - \frac{4}{49} = \frac{45}{49} \][/tex]

Next, take the square root of both sides to solve for [tex]\(\sin(\theta)\)[/tex]:

[tex]\[ \sin(\theta) = \pm \sqrt{\frac{45}{49}} \][/tex]

Simplify the square root:

[tex]\[ \sin(\theta) = \pm \frac{\sqrt{45}}{7} = \pm \frac{3\sqrt{5}}{7} \][/tex]

Since the terminal side of [tex]\(\theta\)[/tex] is in quadrant IV, and sine is negative in this quadrant, we select the negative value:

[tex]\[ \sin(\theta) = -\frac{3\sqrt{5}}{7} \][/tex]

Therefore, the exact, fully simplified and rationalized answer is:

[tex]\[ \sin(\theta) = -\frac{3\sqrt{5}}{7} \][/tex]