given that csc(theta) =-3 and cos(theta)>0, find the remaining 5 trigonometric functions of theta.
i have my triangle graphed in the fourth quadrant but I'm getting confused because of the negatives...
cscØ = -3 , so sinØ = -1/3
the sine is negative in III and IV
cosine is positive in I and IV
so you are correct to say that Ø is in IV
make yourself familiar with the CAST rule, here is an explanation for it
http://www.mathsrevision.net/alevel/pages.php?page=36
So your triangle in IV should be a 1-√8 -3 triangle
sinØ = -1/3 , cscØ = -3 (that was the given)
cos = √8/3 , secØ = 3/√8
tanØ = -1/√8 , cotØ = -√8
(Your textbook answer might have written the √8 as 2√2 )
To find the remaining trigonometric functions of theta, we start with the given information:
csc(theta) = -3 (in the fourth quadrant)
cos(theta) > 0
Since cos(theta) = 1 / sec(theta), we can replace cos(theta) in terms of sec(theta):
sec(theta) = 1 / cos(theta) = 1 / (1 / sec(theta)) = sec(theta)
Using the relationship between sec(theta) and csc(theta), we can use the Pythagorean identity:
sec^2(theta) = csc^2(theta) + 1
Substituting the values, we have:
sec^2(theta) = (-3)^2 + 1 = 9 + 1 = 10
Taking the square root of both sides, we find:
sec(theta) = sqrt(10)
Now, we can find the remaining trigonometric functions:
1. cos(theta):
Since cos(theta) > 0, in the fourth quadrant, cos(theta) is positive:
cos(theta) = sqrt(1 - sin^2(theta)) = sqrt(1 - (-3)^2) = sqrt(1 - 9) = sqrt(-8) (Note that the square root of a negative number is not defined for real numbers, so we cannot determine a numerical value for cos(theta))
2. sin(theta):
Using the definition of csc(theta), we have:
csc(theta) = 1 / sin(theta) = -3
sin(theta) = 1 / csc(theta) = 1 / (-3) = -1/3
3. tan(theta):
Using the definition of tan(theta), we have:
tan(theta) = sin(theta) / cos(theta) = (-1/3) / sqrt(-8) (Note that tan(theta) is undefined since cos(theta) is undefined)
4. cot(theta):
Using the definition of cot(theta), we have:
cot(theta) = 1 / tan(theta) = 1 / (-1/sqrt(8)) (Note that cot(theta) is undefined since tan(theta) is undefined)
5. sec(theta):
We already calculated sec(theta) earlier:
sec(theta) = sqrt(10)
6. csc(theta):
We were given csc(theta) as -3, so csc(theta) = -3.
To summarize:
cos(theta) = undefined
sin(theta) = -1/3
tan(theta) = undefined
cot(theta) = undefined
sec(theta) = sqrt(10)
csc(theta) = -3
To find the remaining trigonometric functions of theta, given that csc(theta) = -3 and cos(theta) > 0, we need to first determine the value of sine (sin) and cosine (cos) using the given information.
We are given that csc(theta) = -3, which is the reciprocal of sine. Therefore, we can write:
sine(theta) = 1 / csc(theta) = 1 / (-3) = -1/3
Next, we know that cos(theta) > 0, which means that cosine is positive in the fourth quadrant. From the given information, we can conclude that:
cosine(theta) = cos(theta) > 0
Now, we can use the values of sine and cosine to find the remaining trigonometric functions:
1. Tangent (tan): tangent(theta) = sin(theta) / cos(theta)
Substitute the values of sin(theta) and cos(theta) into the formula:
tan(theta) = (-1/3) / cos(theta)
2. Cotangent (cot): cot(theta) = 1 / tan(theta)
Use the value of tan(theta) obtained in the previous step to find cot(theta).
3. Secant (sec): sec(theta) = 1 / cos(theta)
Use the positive value of cos(theta) to find sec(theta).
4. Cosecant (csc): csc(theta) = 1 / sin(theta)
Use the value of sin(theta) to find csc(theta).
5. The angle theta is located in the fourth quadrant, where sine and cosine are negative. Therefore, the signs of the remaining trigonometric functions will be as follows:
For tan, cot, sec, and csc:
- tan(theta) = -1/3
- cot(theta) = reciprocal of tan(theta)
- sec(theta) = reciprocal of positive cosine(theta)
- csc(theta) = reciprocal of negative sine(theta)