Question

if secthetha=5/3 and 0<thetha<pi/2. find all the T-ratios​

Answer 1

Given that

0 < ∅ < π/2

This means that the angle ∅ can not be greater than π/2 and can not be smaller than 0

This further means that all trigonometric ratios will be positive.

Now, let's focus on the question.

Sec∅ = 5/3

Taking out cos∅ is the easiest, it's just the reciprocal of sec∅

so, cos∅ = 1/sec∅ = 1÷ (5/3) = 3/5

Now, there's a cool identity that puts sin∅ and cos∅ in a relation. It is

sin²∅ + cos²∅ = 1

Now we know the value of cos∅ so we can find sin∅

sin²∅ + (3/5)² = 1

sin²∅ = 1 - (3/5)²

= 1 - 9/25

= (25 - 9)/25

= 16/25

as sin²∅ = 16/25

sin∅ = √(16/25) = 4/5

(We didn't take the negative value because all Trigonometric ratios are positive. Why? Go back to the starting of this solution.)

Now tan∅ = sin∅/cos∅

so 4/5 ÷ 3/5

= 4/3

Now cosec∅ = 1/sin∅

= 1 ÷ 4/5

= 5/4

And cot∅ = 1/tan∅

= 1 ÷ 4/3

= 3/4

Final answer

sin∅ = 4/5

cos∅ = 3/5

tan∅ = 4/3

sec∅ = 5/3

cosec∅ = 5/4

cot∅ = 3/4