cot(theta)= 3

pi < theta < 3pi/2
Find:
sin(theta)= -1 ?
cos(theta)= -3 ?
tan(theta)= 1/3 ?
sec(theta)= -1/3 ?
csc(theta)= -1 ?

That's what I came up with, but they are not correct, PLEASE help me where I went wrong!!

tan theta= 1/3 which means sintheta 1/sqr10

draw the triangle Notice it is in the third quadrant, where cosine is negative.

You do not seem to have calculated the hypotenuse which you need for sin and cos and sec and csc

sqrt (1^2+3^2) = sqrt 10

So,

tan(theta)=sin(theta)/cos(theta)
cos(theta)=sin(theta)/tan(theta)
=(1/sqrt(10))/(1/3)
= 3/sqrt(10) ??
But its incorrect!! What am I doing wrong?

If sin t=-1/4, find sin t + 6pi

tan theta= 5 divided by 12 and theta is in quadrant 3 what does sin 2 theta equal

To find the values of sin(theta), cos(theta), tan(theta), sec(theta), and csc(theta) given that cot(theta) = 3 and pi < theta < 3pi/2, we need to use trigonometric identities and equations.

First, let's recall the definition of cotangent:

cot(theta) = cos(theta) / sin(theta)

Since we know that cot(theta) = 3, we can rewrite this equation as:

3 = cos(theta) / sin(theta)

Now, let's square both sides to eliminate the fractions:

9 = cos^2(theta) / sin^2(theta)

Next, we'll use the Pythagorean identity to replace sin^2(theta) with 1 - cos^2(theta):

9 = cos^2(theta) / (1 - cos^2(theta))

Multiply both sides by (1 - cos^2(theta)):

9 - 9cos^2(theta) = cos^2(theta)

Rearrange the equation:

10cos^2(theta) - 9 = 0

To solve this quadratic equation, let's use the quadratic formula:

cos(theta) = (-b ± √(b^2 - 4ac)) / 2a

Where a = 10, b = 0, and c = -9.

Substituting these values into the formula, we have:

cos(theta) = (± √(0^2 - 4 * 10 * -9)) / (2 * 10)
cos(theta) = (± √(360)) / 20
cos(theta) = ± √(9 * 4 * 10) / 20
cos(theta) = ± (3 * 2 * √10) / 20
cos(theta) = ± (3 * √10) / 10
cos(theta) = ± (√10) / 10

Given that theta is in the third quadrant (pi < theta < 3pi/2), we know that cos(theta) is negative. Therefore, we can write:

cos(theta) = - (√10) / 10

Now, we can find sin(theta) using the equation:

sin(theta) = cos(theta) / cot(theta)

Substituting the values we have:

sin(theta) = (- (√10) / 10) / 3
sin(theta) = - (√10) / (10 * 3)
sin(theta) = - (√10) / 30

Now let's find tan(theta) using the relationship:

tan(theta) = sin(theta) / cos(theta)

Substituting the values we have:

tan(theta) = (- (√10) / 30) / (- (√10) / 10)
tan(theta) = (√10 / 30) * (10 / √10)
tan(theta) = (10 / 30) * (1)
tan(theta) = 1/3

Next, we can find sec(theta) using the identity:

sec(theta) = 1 / cos(theta)

Substituting the values we have:

sec(theta) = 1 / (- (√10) / 10)
sec(theta) = -10 / √10
sec(theta) = - (10√10) / 10
sec(theta) = - √10

Finally, we can find csc(theta) using the identity:

csc(theta) = 1 / sin(theta)

Substituting the values we have:

csc(theta) = 1 / (- (√10) / 30)
csc(theta) = -30 / √10
csc(theta) = - (30√10) / 10
csc(theta) = - 3√10

Therefore, the correct values are:
sin(theta) = - (√10) / 30
cos(theta) = - (√10) / 10
tan(theta) = 1/3
sec(theta) = - √10
csc(theta) = - 3√10

I hope this explanation helps you find the correct answers!