Question

find the values of the six trigonometric functions for angle theta in standard position if a point with the coordinates (1, -8) lies on its terminal side

Answer 1

cosФ =
`(1)/(√(65))` , sinФ =
`-(8)/(√(65))` , tanФ = -8, secФ =
`√(65)` , cscФ =
`-(√(65))/(8)` , cotФ =
`-(1)/(8)`

Explanation:

If a point (x, y) lies on the terminal side of angle Ф in standard position, then the six trigonometry functions are:

1. cosФ =
   `(x)/(r)`
2. sinФ =
   `(y)/(r)`
3. tanФ =
   `(y)/(x)`
4. secФ =
   `(r)/(x)`
5. cscФ =
   `(r)/(y)`
6. cotФ =
   `(x)/(y)`

• Where r =
  `\sqrt{x^(2)+y^(2) }` (the length of the terminal side from the origin to point (x, y)

• You should find the quadrant of (x, y) to adjust the sign of each function

∵ Point (1, -8) lies on the terminal side of angle Ф in standard position

∵ x is positive and y is negative

→ That means the point lies on the 4th quadrant

∴ Angle Ф is on the 4th quadrant

∵ In the 4th quadrant cosФ and secФ only have positive values

∴ sinФ, secФ, tanФ, and cotФ have negative values

→ let us find r

∵ r =
`\sqrt{x^(2)+y^(2) }`

∵ x = 1 and y = -8

∴ r =
`√(x) \sqrt{(1)^(2)+(-8)^(2)}=√(1+64)=√(65)`

→ Use the rules above to find the six trigonometric functions of Ф

∵ cosФ =
`(x)/(r)`

∴ cosФ =
`(1)/(√(65))`

∵ sinФ =
`(y)/(r)`

∴ sinФ =
`-(8)/(√(65))`

∵ tanФ =
`(y)/(x)`

∴ tanФ =
`-(8)/(1)` = -8

∵ secФ =
`(r)/(x)`

∴ secФ =
`(√(65))/(1)` =
`√(65)`

∵ cscФ =
`(r)/(y)`

∴ cscФ =
`-(√(65))/(8)`

∵ cotФ =
`(x)/(y)`

∴ cotФ =
`-(1)/(8)`