Question

If sin of theta equals 3/8 and theta is in quadrant II. what are cos, tan, csc, cot, and sec of theta?

Answer 1

See explanation below.

Explanation:

`sintheta = 3/8`, `theta` in quadrant `II`

Imagine a right triangle being drawn on the cartesian plane, as in the following example.

Since sin = opposite/hypotenuse, the side opposite to `theta` is 3 and the hypotenuse is `8`, we can rearrange our pythagorean theorem to find the adjacent side, b.

`a^2 + b^2 = c^2`

`b^2 = c^2 - a^2`

`b^2 = 8^2 - 3^2`

`b^2 = 64 - 9`

`b^2 = 55`

`b = -sqrt(55)`

So, now we know that the adjacent side measures `-sqrt(55)` units (since the x axis is negative in quadrant II) in length. Thus we can deduce that `costheta = -sqrt(55)/8` and `tantheta = -3/(sqrt(55))`

Now, for `csctheta, sectheta and cottheta`, we must apply the reciprocal identities.

`csctheta = 1/sintheta`

`sectheta = 1/costheta`

`cottheta = 1/tantheta`

Therefore, `csctheta = 8/3, sectheta = -8/sqrt(55) and cottheta = -sqrt(55)/3`

To summarize, the six trigonometric ratios, we get:

`sintheta = 3/8`

`costheta = -sqrt(55)/8`

`tantheta = -3/sqrt(55)`

`csctheta = 8/3`

`sectheta = -8/sqrt(55)`

`cottheta = -sqrt(55)/3`

Hopefully this helps!

Answer 2

`"see explanation"`

Explanation:

`sintheta=3/8to(color(red)(1))`

`• csctheta=1/sintheta=8/3to(color(red)(2))`

`• costheta=+-sqrt(1-sin^2theta)`

`color(white)(xxxxx)=-sqrt(1-9/64)larr" negative value"`

`color(white)(xxxxx)=-sqrt(55/64)`

`rArrcostheta=-sqrt55/8to(color(red)(3))`

`• sectheta=1/costheta=-8/sqrt55to(color(red)(4))`

`• tantheta=(sintheta)/(costheta)`

`color(white)(xxxx)=3/8xx-8/sqrt55=-3/sqrt55to(color(red)(5))`

`• cottheta=1/tantheta=-sqrt55/3to(color(red)(6))`