Question

Assume that the terminal side of thetaθ passes through the point (negative 12 comma 5 )(−12,5) and find the values of trigonometric ratios sec thetaθ and sin thetaθ.

Answer 1

`\sin \theta = (5)/(13)` and
`\sec \theta = -(13)/(12)`

Explanation:

Assume that the terminal side of thetaθ passes through the point (−12,5).

In ordered pair (-12,5), x-intercept is negative and y-intercept is positive. It means the point lies in 2nd quadrant.

Using Pythagoras theorem:

`hypotenuse^2=perpendicular^2+base^2`

`hypotenuse^2=(5)^2+(12)^2`

`hypotenuse^2=25+144`

`hypotenuse^2=169`

Taking square root on both sides.

`hypotenuse=13`

In a right angled triangle

`\sin \theta = (opposite)/(hypotenuse)`

`\sin \theta = (5)/(13)`

`\sec \theta = (hypotenuse)/(adjacent)`

`\sec \theta = (13)/(12)`

In second quadrant only sine and cosecant are positive.

`\sin \theta = (5)/(13)` and
`\sec \theta = -(13)/(12)`