Question

If `sin θ = 1/3`, find the values of other t-ratios.

Answer 1

1. Identify given sides

The sine ratio is defined as the ratio of the opposite side to the hypotenuse in a right- angled triangle.

`sin θ = "Opposite"/"Hypotenuse"`

= `1/3`

Thus, we can let:

Opposite = 1

Hypotenuse = 3

2. Calculate the adjacent side

 Using the Pythagoras theorem (H² = P² + B² or hyp² = opp² + adj²), we find the base (adjacent side):

3² = 1² + Adjacent²

9 = 1 + Adjacent²

Adjacent² = 8

Adjacent = `sqrt(8)`

Adjacent = `2sqrt(2)`

3. Determine other ratios

 Now, substitute the values of the opposite (1), hypotenuse (3), and adjacent `(2sqrt(2))` into the standard trigonometric formulas:

`cos θ = "Adjacent"/"Hypotenuse"`

= `(2sqrt(2))/3`

`tan θ = "Opposite"/"Adjacent"`

= `1/(2sqrt(2))` (or `sqrt(2)/4` when rationalised)

`"cosec"  θ = 1/(sin θ)`

= 3

`sec θ = 1/(cos θ)`

= `3/(2sqrt(2))` (or `(3sqrt(2))/4` when rationalised)

`cot θ = 1/(tan θ)`

= `2sqrt(2)`