Which of the following is not a possible solution of Sine of theta

Question

Which of the following is not a possible solution of ?

	A.
	B.
	C.
	D.

Solve the equation  for . 

	A.
	B.
	C.
	D.

 

Answer 1

Equation : sinθ + cosθtan²θ =0

Apply quotient rule : tan θ = sin(θ)/cos(θ)

sinθ + cosθ (sinθ/cosθ)² =0

sinθ + sin²θ/cosθ =0

sinθcos θ+ sin²θ =0

sinθ(cos θ+ sinθ) =0

sinθ = 0

sinθ  = sin0

General solution : If sin(θ) = sin(α) then θ = nπ + (-1)ⁿ α, where n is integer.

If α = 0 then angle θ = nπ

The solutions are π,2π, 3π........

cos θ+ sinθ=0

sinθ = - cos θ

tan θ =-1

θ = - π/4

General solution : If tan(θ) = tan(α) then θ = nπ + α, where n is integer.

If α = - π/4 then angle θ = nπ - π/4

The solutions are 3π/4,7π/4,.......

So 5π/2 is not a solution to given equation.

Option A is correct.

Answer 2

Equation : cot²(π/2 - θ) =secθ +1

Where cot(π/2 - θ) = tan θ

tan²(θ) =secθ +1

sec²(θ) -1 =secθ +1            (tan²(θ) = sec²(θ) -1)

(sec²(θ) -1) - (secθ +1) =0

(sec(θ) -1) (secθ +1)- (secθ +1) =0

(secθ +1)(sec(θ) -2) =0

secθ +1 =0

secθ = -1
cos θ =-1

cos θ  =cos π

General solution : If cos(θ) = cos(∝) then θ = 2nπ ± ∝, where n is integer.

If ∝ = π then angleθ = 2nπ ± π = (2n ±1)π

The solutions are π

secθ -2 =0

secθ = 2
cos θ =1/2

cos θ  =cos π/3

General solution : If cos(θ) = cos(∝) then θ = 2nπ ± ∝, where n is integer.

If ∝ = π/3 then angleθ = 2nπ ± π/3

The solutions are π/3,5π/3.

Option C is correct.