Final answer:
To find sin(x/2), cos(x/2), and tan(x/2), we can use the half-angle identities. For the given information cot(x) = 5, we can find sin(x) and cos(x), and then use the half-angle identities to find sin(x/2), cos(x/2), and tan(x/2). So, sin(x/2) = ±√((26 - 5√26)/(26 + 5√26)), cos(x/2) = ±√((26 + 5√26)/52), and tan(x/2) = ±√((26 - 5√26)/(26 + 5√26)).
Step-by-step explanation:
To find sin(x/2), cos(x/2), and tan(x/2) from the given information, we need to use the half-angle identities. Let's start by finding the values of sin(x), cos(x), and tan(x) using the given information. Given that cot(x) = 5, we can write it as cos(x)/sin(x) = 5. From this equation, we can deduce that cos(x) = 5sin(x). We also know that cos²(x) + sin²(x) = 1, so substituting for cos(x), we get (5sin(x))² + sin²(x) = 1. Simplifying this equation, we get 26sin²(x) = 1, which gives sin(x) = ±√(1/26). Since x lies in quadrant II (180°), sin(x) is positive. Therefore, sin(x) = √(1/26).
To find the value of sin(x/2), we can use the half-angle identity for sine: sin(x/2) = ±√((1 - cos(x))/2). Substituting the value of cos(x) that we found earlier, we get sin(x/2) = ±√((1 - 5sin(x))/2). Plugging in the value of sin(x), we get sin(x/2) = ±√((1 - 5√(1/26))/2). Simplifying further, we get sin(x/2) = ±√((26 - 5√26)/52).
Similarly, we can find the value of cos(x/2) using the half-angle identity for cosine: cos(x/2) = ±√((1 + cos(x))/2). Substituting the value of cos(x), we get cos(x/2) = ±√((1 + 5sin(x))/2). Plug in the value of sin(x), we get cos(x/2) = ±√((1 + 5√(1/26))/2). Simplifying further, we get cos(x/2) = ±√((26 + 5√26)/52).
We can also find tan(x/2) using the half-angle identity for tangent: tan(x/2) = sin(x/2)/cos(x/2). Substituting the values of sin(x/2) and cos(x/2), we get tan(x/2) = ±√((26 - 5√26)/(26 + 5√26)).