Find the exact value of cotθ given secθ=3 and tanθ<0.
cotθ=8√/3
cotθ=−(3/√8)
cotθ=1/√8
cotθ=−(√8/3)
cotθ=−√8
cotθ=3/√8
cotθ=−(1/√8)
cotθ=√8
Is it cotθ=−(1/√8)?
thanks
cos =1/3 so positive x
cos/sin = negative so y is negative, quadrant 4
sin = - sqrt 8 / 3
cot = cos/sin= (1/3) / (- sqrt 8 /3) = -1 / sqrt 8
so yes
by the way sqrt 8 = 2 sqrt 2
correct
To determine the exact value of cotθ, given secθ=3 and tanθ<0, we can use the relationships between trigonometric functions. Here's how you can find the answer:
1. Start by recalling the definitions of the trigonometric functions:
- secθ = 1/cosθ
- cotθ = 1/tanθ
2. Since secθ=3, we can find the value of cosθ by taking the reciprocal:
cosθ = 1/secθ
= 1/3
3. Next, we need to determine the sign of sinθ. Since tanθ<0, it means that either sinθ or cosθ is negative. In this case, we know that cosθ is positive (since secθ=3), so sinθ must be negative.
4. Using the Pythagorean identity, we can find sinθ:
sinθ = -√(1 - cos²θ)
= -√(1 - (1/3)²)
= -√(1 - 1/9)
= -√(8/9)
= -√8 / √9
= -√8 / 3
5. Now that we have the values of cosθ and sinθ, we can find the value of tanθ:
tanθ = sinθ / cosθ
= (-√8 / 3) / (1/3)
= -√8 / 1
= -√8
6. Finally, we can find the value of cotθ by taking the reciprocal of tanθ:
cotθ = 1 / tanθ
= 1 / (-√8)
= -1 / √8
= -(1/√8)
= -√8/8
So, the correct value of cotθ is cotθ=-(√8/8), not cotθ=-(1/√8).
Hope this explanation helps! Let me know if you have any more questions.