cotx=-3/5,-pi/2<x<0
Use identities to find the exact values for the remaining five trigonometric functions
We use the fact that $\cot x = \frac{1}{\tan x}$ and $\tan^2 x + 1 = \sec^2 x$.
Since $\cot x = -\frac{3}{5}$ and $-\frac{\pi}{2} < x < 0$, we know that $\tan x = -\frac{5}{3}$ (since $\tan x$ is negative in the second quadrant). Thus, we have
$$\sec^2 x = \tan^2 x + 1 = \left(-\frac{5}{3}\right)^2 + 1 = \frac{34}{9}$$
Taking the square root, we get $\sec x = \pm \frac{\sqrt{34}}{3}$. Since $-\frac{\pi}{2} < x < 0$, we know that $\cos x < 0$, so $\sec x$ is negative. Therefore, we have $\sec x = -\frac{\sqrt{34}}{3}$.
Now we can find the other trigonometric functions:
$$\begin{aligned} \sin x &= \frac{\tan x}{\sec x} = -\frac{5}{\sqrt{34}} \\ \cos x &= -\frac{1}{\sec x} = \frac{\sqrt{34}}{3} \\ \csc x &= \frac{1}{\sin x} = -\frac{\sqrt{34}}{5} \\ \sec x &= -\frac{\sqrt{34}}{3} \\ \cot x &= -\frac{3}{5} \end{aligned}$$
Therefore, the exact values of the six trigonometric functions are:
$$\begin{aligned} \sin x &= -\frac{5}{\sqrt{34}}, \\ \cos x &= \frac{\sqrt{34}}{3}, \\ \tan x &= -\frac{5}{3}, \\ \csc x &= -\frac{\sqrt{34}}{5}, \\ \sec x &= -\frac{\sqrt{34}}{3}, \\ \cot x &= -\frac{3}{5}. \end{aligned}$$
To find the exact values for the remaining five trigonometric functions, we will use the given equation cot(x) = -3/5 and the Pythagorean identity, sin^2(x) + cos^2(x) = 1.
Step 1: Find the value of sin(x).
Since cot(x) = cos(x)/sin(x), we can write the given equation as cos(x)/sin(x) = -3/5.
Cross-multiplying, we get cos(x) = -3sin(x)/5.
Using the Pythagorean identity, sin^2(x) = 1 - cos^2(x), we can substitute the value of cos(x) we found:
sin^2(x) = 1 - (-3sin(x)/5)^2.
Expanding the square, we get:
sin^2(x) = 1 - 9sin^2(x)/25.
Multiplying both sides by 25, we have:
25sin^2(x) = 25 - 9sin^2(x).
Combining like terms, we get:
34sin^2(x) = 25.
Dividing both sides by 34, we get:
sin^2(x) = 25/34.
Taking the square root of both sides, we find:
sin(x) = ±√(25/34) = ±5√(2/34)= ±5√2/√34 = ±5√2/(√2√17) = ±5/√17 = ±5√17/17.
Step 2: Find the value of cos(x).
Using the equation cos(x) = -3sin(x)/5, we can substitute the value of sin(x) we found:
cos(x) = -3(±5√17/17)/5.
Simplifying, we get:
cos(x) = -(±3√17)/17.
Step 3: Find the value of tan(x).
Using the identity tan(x) = sin(x)/cos(x), we can substitute the values of sin(x) and cos(x) we found:
tan(x) = (±5√17/17)/(-(±3√17)/17).
Simplifying, we get:
tan(x) = (±5√17/17) / (±3√17/17).
Dividing both the numerator and denominator by 17, we have:
tan(x) = (±5√17) / (±3√17).
Simplifying further, we get:
tan(x) = ±5/3.
Step 4: Find the value of sec(x) using the identity sec(x) = 1/cos(x).
Substituting the value of cos(x) we found, we get:
sec(x) = 1/(-(±3√17)/17).
Simplifying, we have:
sec(x) = -17/(±3√17).
Therefore, sec(x) = -17/(3√17) or sec(x) = 17/(3√17).
Step 5: Find the value of csc(x) using the identity csc(x) = 1/sin(x).
Substituting the value of sin(x) we found, we get:
csc(x) = 1/(±5√17/17).
Simplifying, we have:
csc(x) = 17/(±5√17).
Therefore, csc(x) = 17/(5√17) or csc(x) = -17/(5√17).
Therefore, the exact values for the remaining five trigonometric functions are:
sin(x) = ±5√17/17
cos(x) = -(±3√17)/17
tan(x) = ±5/3
sec(x) = -17/(3√17) or sec(x) = 17/(3√17)
csc(x) = 17/(5√17) or csc(x) = -17/(5√17)