Looking for answers? Westonci.ca is your go-to Q&A platform, offering quick, trustworthy responses from a community of experts. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
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].
### 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].
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.