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Given that [tex] \sin x = \frac{3}{5} [/tex] and [tex] \cos y = \frac{7}{25} [/tex], with [tex]\(x\)[/tex] and [tex]\(y\)[/tex] both being Quadrant I angles, find [tex] \tan \left( \frac{x}{2} \right) [/tex].

Sagot :

Certainly! Let's tackle this question step-by-step, making use of the given information.

### Step 1: Calculate cos(x)
Given:
[tex]\[ \sin(x) = \frac{3}{5} \][/tex]

First, understand that for any angle [tex]\( x \)[/tex]:
[tex]\[ \sin^2(x) + \cos^2(x) = 1 \][/tex]
So, we can find [tex]\(\cos(x)\)[/tex] by rearranging this equation. Since [tex]\( x \)[/tex] is in the first quadrant, [tex]\(\cos(x)\)[/tex] is positive:

[tex]\[ \cos^2(x) = 1 - \sin^2(x) \][/tex]

[tex]\[ \cos^2(x) = 1 - \left(\frac{3}{5}\right)^2 \][/tex]

[tex]\[ \cos^2(x) = 1 - \frac{9}{25} \][/tex]

[tex]\[ \cos^2(x) = \frac{25}{25} - \frac{9}{25} \][/tex]

[tex]\[ \cos^2(x) = \frac{16}{25} \][/tex]

Thus:
[tex]\[ \cos(x) = \sqrt{\frac{16}{25}} = \frac{4}{5} \][/tex]

### Step 2: Calculate tan(x)
Next, we find [tex]\(\tan(x)\)[/tex] using [tex]\(\sin(x)\)[/tex] and [tex]\(\cos(x)\)[/tex]:

[tex]\[ \tan(x) = \frac{\sin(x)}{\cos(x)} \][/tex]

[tex]\[ \tan(x) = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \][/tex]

### Step 3: Calculate tan(x/2) using the half-angle identity
For the tangent of a half-angle, we use the half-angle identity:

[tex]\[ \tan\left(\frac{x}{2}\right) = \sqrt{\frac{1 - \cos(x)}{1 + \cos(x)}} \][/tex]

Since [tex]\( x \)[/tex] is in the first quadrant, [tex]\( \tan\left(\frac{x}{2}\right) \)[/tex] will be positive:

[tex]\[ \tan\left(\frac{x}{2}\right) = \sqrt{\frac{1 - \frac{4}{5}}{1 + \frac{4}{5}}} \][/tex]

[tex]\[ \tan\left(\frac{x}{2}\right) = \sqrt{\frac{\frac{1}{5}}{\frac{9}{5}}} \][/tex]

[tex]\[ \tan\left(\frac{x}{2}\right) = \sqrt{\frac{1}{9}} = \frac{1}{3} \][/tex]

Thus, the value of [tex]\(\tan\left(\frac{x}{2}\right)\)[/tex] is:

[tex]\[ \tan\left(\frac{x}{2}\right) = \frac{1}{3} \][/tex]

### Result
[tex]\[ \tan\left(\frac{x}{2}\right) \approx 0.33333333333333326 \][/tex]

This detailed step-by-step solution finds that the tangent of the half-angle, [tex]\(\tan\left(\frac{x}{2}\right)\)[/tex], is approximately [tex]\(0.33333333333333326\)[/tex].