Question

If `sin θ = 1/sqrt(2)`, find the values of other t-ratios.

Answer 1

1. Identify given values

From the definition of sine in a right-angled triangle,

`sin θ = "Perpendicular (P)"/"Hypotenuse (H)"`

Perpendicular (P) = 1

Hypotenuse (H) = `sqrt(2)`

2. Find the adjacent side

Using the Pythagoras theorem (H² = P² + B²), we calculate the Base (B):

`(sqrt(2))^2 = (1)^2 + B^2`

2 = 1 + B²

B² = 1

⇒ B = 1

3. Calculate all t-ratios

Using the values P = 1, B = 1 and H = `sqrt(2)`, we find the remaining ratios:

`cos θ: "Base"/"Hypotensuse"`

= `1/sqrt(2)`

`tan θ: "Perpendicular"/"Base"`

= `1/1`

= 1

`"cosec"  θ: 1/(sin θ)`

= `sqrt(2)/1`

= `sqrt(2)`

`sec θ: 1/(cos θ)`

= `sqrt(2)/1`

= `sqrt(2)`

`cot θ: 1/(tan θ)`

= `1/1`

= 1