Find the remaining trigonometric ratios:
tan(x)=2, 0 < x < pi/2
1. sin(x)= _____
2. csc(x)= _____
3. cos(x)= _____
4. sec(x)= _____
wrong again.
not quite ....
sinx = 2/√5 or 2√5/5
cscx = √5/2
cosx = 1/√5 or √5/5
secx = √5
cotx = 1/2
To find the remaining trigonometric ratios when tan(x) = 2 and 0 < x < pi/2, we can use the following relationships:
1. sin(x) = tan(x) * cos(x)
2. csc(x) = 1 / sin(x)
3. cos(x) = 1 / sec(x)
4. sec(x) = 1 / cos(x)
Let's calculate each ratio step by step:
1. sin(x) = tan(x) * cos(x)
Since tan(x) = 2, we need to find cos(x) to calculate sin(x).
Using the Pythagorean identity, sin^2(x) + cos^2(x) = 1, we get:
(2)^2 + cos^2(x) = 1
4 + cos^2(x) = 1
cos^2(x) = 1 - 4
cos^2(x) = -3 (which is not possible)
Hence, sin(x) is not defined for the given conditions.
2. csc(x) = 1 / sin(x)
Since sin(x) is not defined, csc(x) is also not defined for the given conditions.
3. cos(x) = 1 / sec(x)
To find cos(x), we need sec(x). Let's calculate sec(x) first.
sec(x) = 1 / cos(x)
Since we don't have the value of cos(x), we cannot find sec(x) for the given conditions.
4. sec(x) = 1 / cos(x)
We don't have the value of cos(x), so sec(x) is also not defined for the given conditions.
Therefore, sin(x), csc(x), cos(x), and sec(x) are all not defined for the given conditions when tan(x) = 2 and 0 < x < pi/2.
5. cot(x)= _____
1. sin(x)= √(1/5)
2. csc(x)= √(5)
3. cos(x)= √(4/5)
4. sec(x)= √(5/4)
5. cot(x)= 1/2