Question

root 3 sin theta - cos theta = zero then find the value of sin squared theta minus cos squared theta

Answer 1

Step-by-step explanation:

Given:-

√3 sinθ - cosθ = 0

To find:-

Find the value of sin^2 θ - cos^2 θ ?

Solution:-

Given equation is √3 sinθ - cosθ = 0

=>√3 sinθ = cosθ

=>√3 = cosθ/sinθ

=>√3 = Cotθ

=>1/√3 = 1/Cotθ

=>1/√3 = tanθ

=>tanθ = 1/√3

=>tanθ = tan 30°

=> θ = 30°

The value of θ = 30°

Now , the value of sin^2 θ - cos^2 θ

=>(sin 30)^2 -( cos 30)^2

=>(1/2)^2 - (√3/2)^2

=>(1/4) - (3/4)

=>(1-3)/4

=>-2/4

=>-1/2

(or)

The value of sin^2 θ - cos^2 θ

=>-(-sin^2 θ +cos^2 θ)

=>-(cos^2θ -sin^2θ )

=>-(cos2θ )

=> -(cos(2×30°))

=>-(cos60°)

=>-(1/2)

=>-1/2

Answer:-

The value of sin^2 θ - cos^2 θ = -1/2

Check:-

Put θ = 30° in the LHS in the given equation√3 sinθ - cosθ

=>√3 sin30° - cos 30°

=>√3(1/2) - (√3/2)

=>(√3/2)-(√3/2)

=>(√3-√3)/2

=>0/2

=>0

=>RHS

LHS = RHS

Used formulae:-

• Cotθ = 1/Tanθ
• Tan 30° = 1/√3
• cos^2θ -sin^2θ = cos2θ
• cos60° = 1/2