Question

cosec theta equals to √2,then find other five trigonometric ratios and then find the value of ratios. ​

Answer 1

Step-by-step explanation:

Given :-

Cosec θ = √2

To find :-

Find the other trigonometric ratios ?

Solution :-

Method -1 :-

Given that

Cosec θ = √2

=> Hypotenuse / Opposite side to θ = √2

=> Hypotenuse / Opposite side to θ = √2/1

Let Hypotenuse = √2 k

Opposite side = 1k = k

Consider a right angled triangle ABC ,

We have , angle B = 90° , angle A = θ

and BC = k and AC = √2 k

By Pythagoras Theorem,

=> AC² = AB²+BC²

=> AB² = AC² - BC²

=> AB² = (√2k)²-(k)²

=> AB² = 2k²-k²

=> AB² = k²

=> AB =√k²

=> AB = k

Therefore, Adjacent side to θ = k

I) Sin θ = Opposite side to θ /Hypotenuse

=> Sin θ = BC/ AC

=> Sin θ = k/ √2 k

Therefore, Sin θ = 1/√2

ii) Cos θ=Adajacent side to θ/Hypotenuse

=> Cos θ = AB/ AC

=> Cos θ = k/ √2 k

Therefore, Cos θ = 1/√2

iii) Tan θ = Opposite side to θ/ Adjacent side to θ

=> Tan θ = BC/AB

=> Tan θ = 1k/1k

Therefore Tan θ = 1

iv) Sec θ =Hypotenuse/Adjacent side to θ

=> Sec θ = AC/ AB

=> Sec θ = √2k/k

Therefore, Sec θ = √2 = 1.4 units

v) Cot θ = Adjacent side to θ/Opposite side to θ

=> Cot θ = AB/BC

=> Cot θ = k/k

Therefore, Cot θ = 1

Method-2:-

Given that

Cosec θ =√2

1/Sin θ = √2

=> Sin θ = 1/√2 = 0.7 units

On squaring both sides then

=> Sin² θ = (1/√2)²

=> Sin² θ = 1/2

=> 1-Sin² θ = 1-(1/2)

=> 1-Sin² θ= 1/2

=> Cos² θ = 1/2

=> Cos θ = √(1/2)

=> Cos θ = 1/√2

=> Cos θ = (1/√2)×(√2/√2)

=> Cos θ = √2/(√2×√2)

=> Cos θ = √2/2

=> Cos θ = 1.414.../2

=> Cos θ = 0.702...

=> Cos θ = 0.7 units

We know that

Tan θ = Sin θ/ Cos θ

=> Tan θ =( 1/√2)/(1/√2)

Tan θ = 1

We know that

Cot θ = 1/ Tan θ

=> Cot θ = 1/1

Cot θ = 1

Cos θ = 1/√2

We know that

Sec θ = 1/Cos θ

=> Sec θ = 1/(1/√2)

Sec θ = √2

=> Sec θ = 1.414...

Sec θ = 1.4 units

Answer:-

i)Sin θ = 1/√2 = 0.7 units

ii)Cos θ = 1/√2 = 0.7 units

ii)Tan θ = 1

iv)Cot θ = 1

v)Sec θ = √2 = 1.4 units

Used formulae:-

Sin θ = Opposite side to θ /Hypotenuse

Cos θ = Adajacent side to θ / Hypotenuse

Tan θ=Opposite side to θ/Adjacent side to θ

Cot θ = Adjacent side to θ/Opposite side to θ

Sec θ =Hypotenuse/Adjacent side to θ

Cosec θ =Hypotenuse/Opposite side to θ

Cosec θ = 1 / Sin θ

Sec θ = 1 / Cos θ

Tan θ = Sin θ / Cos θ

Cot θ = 1 / Tan θ

Sin² A + Cos² A = 1

Pythagoras Theorem:-

In a right angled triangle The square of the hypotenuse is equal to the sum of the other two sides .

√2 = 1.414...