Question

Find the exact values of the five remaining trigonometric functions of theta.

21. sec theta = √3, where sin theta 0​

Answer 1

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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