Question

Let (8,−3) be a point on the terminal side of θ. Find the exact values of cosθ, cscθ, and tanθ.

Answer 1

`\text{Cos}\theta=\frac{\text{Adjacent side}}{\text{Hypotenuse}}=(x)/(R)=(8)/(√(73))`

`\text{Csc}\theta=-(√(73))/(3)`

`\text{tan}\theta =\frac{\text{Opposite side}}{\text{Adjacent side}}=(y)/(x)=(-3)/(8)`

Explanation:

From the picture attached,

(8, -3) is a point on the terminal side of angle θ.

Therefore, distance 'R' from the origin will be,

R =
`\sqrt{x^(2)+y^(2)}`

R =
`\sqrt{8^(2)+(-3)^2}`

=
`√(64+9)`

=
`√(73)`

Therefore, Cosθ =
`\frac{\text{Adjacent side}}{\text{Hypotenuse}}=(x)/(R)=(8)/(√(73))`

Sinθ =
`\frac{\text{Opposite side}}{\text{Hypotenuse}}=(y)/(R)=(-3)/(√(73) )`

tanθ =
`\frac{\text{Opposite side}}{\text{Adjacent side}}=(y)/(x)=(-3)/(8)`

Cscθ =
`\frac{1}{\text{Sin}\theta}=(R)/(y)=-(√(73))/(3)`