Let csc(x)=-3. Find all possible values of sin(x)+cos(x)
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sec(x)
Well, first of all, I have to say that csc(x)=-3 sounds like a pretty negative relationship. I guess x must have done something to upset csc.
Now, let's tackle the math problem. We know that csc(x) is equal to -3. Cosecant is the reciprocal of sine, so we can rewrite this as sin(x) = -1/3.
We can also recall that sec(x) is the reciprocal of cosine, so we can rewrite the expression as sin(x) + cos(x) divided by 1/cos(x). Simplifying that, we get sin(x)cos(x) + cos^2(x).
Since we know that sin(x) = -1/3, we can substitute that into the expression and get (-1/3)cos(x) + cos^2(x).
Now, to find the possible values of this expression, it would be helpful to know the value of cos(x).
However, without more information about the domain of x or any other equations related to x, it's impossible to determine the specific values of sin(x) + cos(x)/sec(x). So, I'm sorry to leave you hanging, but humor can't solve everything!
To find possible values of sin(x)+cos(x) / sec(x) when csc(x) = -3, let's start by finding the values of sin(x), cos(x), and sec(x) using the given information.
We are given that csc(x) = -3.
Recall that csc(x) is the reciprocal of sin(x), so we can write the equation as:
1/sin(x) = -3
To find sin(x), we can take the reciprocal of both sides:
sin(x) = 1/(-3)
sin(x) = -1/3
Now, let's find the values of cos(x) and sec(x).
We know that sin^2(x) + cos^2(x) = 1.
Substituting sin(x) = -1/3 into the equation, we have:
(-1/3)^2 + cos^2(x) = 1
1/9 + cos^2(x) = 1
cos^2(x) = 1 - 1/9
cos^2(x) = 8/9
Taking the square root of both sides, we get:
cos(x) = ±√(8/9)
cos(x) = ±(2√2 / 3)
Now, let's find sec(x) using the definition: sec(x) = 1/cos(x).
sec(x) = 1 / (±2√2 / 3)
sec(x) = 3 / (±2√2)
At this point, we have the values of sin(x) = -1/3, cos(x) = ±(2√2 / 3), and sec(x) = 3 / (±2√2).
Finally, we can substitute these values into sin(x) + cos(x) / sec(x):
sin(x) + cos(x) / sec(x) = (-1/3) + (±(2√2 / 3)) / (3 / (±2√2))
Using the reciprocal property, we can rewrite the expression as:
sin(x) + cos(x) / sec(x) = (-1/3) + (±(2√2 / 3)) * (±(2√2) / 3)
simplifying further...
sin(x) + cos(x) / sec(x) = (-1/3) + 4/9
sin(x) + cos(x) / sec(x) = (-3/9) + (4/9)
sin(x) + cos(x) / sec(x) = 1/9
Therefore, the possible values of sin(x) + cos(x) / sec(x) when csc(x) = -3 are 1/9.
To find the possible values of `sin(x) + cos(x) / sec(x)` when `csc(x) = -3`, we need to use the trigonometric identities and equations to determine the values of `x`. Let's break down the steps:
1. Given `csc(x) = -3`, we know that `csc(x)` is the reciprocal of `sin(x)`. So, we have `sin(x) = -1 / 3`.
2. Next, we need to find `cos(x)` and `sec(x)` in terms of `sin(x)`. We can use the Pythagorean identity, `sin^2(x) + cos^2(x) = 1`, to find `cos(x)`:
`sin^2(x) + cos^2(x) = 1`
`(-1/3)^2 + cos^2(x) = 1`
`1/9 + cos^2(x) = 1`
`cos^2(x) = 1 - 1/9`
`cos^2(x) = 8/9`
`cos(x) = √(8/9)`
Similarly, we know that `sec^2(x) = 1 + tan^2(x)`. Since `sec(x)` is the reciprocal of `cos(x)`, we can find `sec(x)`:
`cos(x) = √(8/9)`
`sec(x) = 1 / √(8/9) = √(9/8)`
3. Now that we have the values of `sin(x)`, `cos(x)`, and `sec(x)`, we can substitute them into the expression `sin(x) + cos(x) / sec(x)`:
`sin(x) = -1 / 3`
`cos(x) = √(8/9)`
`sec(x) = √(9/8)`
`sin(x) + cos(x) / sec(x) = (-1 / 3) + (√(8/9)) / (√(9/8))`
`= (-1 / 3) + (√(8/9)) * (√(8/9)) / (√(9/8))`
`= (-1 / 3) + (8/9) / (√(9/8))`
`= (-1 / 3) + (8/9) * (√(8/9) / √(9/8))`
`= (-1 / 3) + (8/9) * (√(8/9) / (3/2))`
`= (-1 / 3) + (8/9) * (2√(8/9) / 3)`
`= (-1 / 3) + (16/27)√(8/9)`
So, the possible values of `sin(x) + cos(x) / sec(x)` when `csc(x) = -3` are `-1/3 + (16/27)√(8/9)`.