Answer:
The secant function is defined as the reciprocal of the cosine function. So if `sec(theta) = √3`, then `cos(theta) = 1/√3`. Since `sin^2(theta) + cos^2(theta) = 1`, we can solve for `sin(theta)`:
`sin^2(theta) + cos^2(theta) = 1`
`sin^2(theta) + (1/√3)^2 = 1`
`sin^2(theta) + 1/3 = 1`
`sin^2(theta) = 2/3`
`sin(theta) = ±√(2/3)`.
Since it is given that `sin theta > 0`, we can conclude that `sin(theta) = √(2/3)`.
Now that we have the values of sine and cosine, we can find the remaining trigonometric functions:
`tan(theta) = sin(theta)/cos(theta)`
`= (√(2/3)) / (1/√3)`
`= √(6)/3`
`cot(theta) = cos(theta)/sin(theta)`
`= (1/√3)/(√(2/3))`
`= √6 / 6`
`csc theta = 1/sin theta`
`= 1/(√(2/3))`
`= √6 / √4`
So, in summary:
- sec theta = √3
- cos theta = 1 / √3
- sin theta = √(2 / 3)
- tan theta = √6 / 3
- cot theta = √6 / 6
- csc theta = √6 / √4
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