Answer:
Step-by-step explanation:
Given the expression;
secθ − tan θ = cosθ
We are to find the solution in the interval [0, 2π).This is as shown;
From trigonometry identity;
secθ = 1/cosθ
tanθ = sinθ/cosθ
Substitute into the formula;
secθ − tan θ = cosθ
1/cosθ-sinθ/cosθ = cosθ
Multiply through by cosθ
1 - sinθ = cos²θ
1-sinθ = (1-sin²θ)
1-sin²θ-1+sinθ =0
-sin²θ+sinθ = 0
sin²θ = sinθ
sinθ = 1
θ = arcsin 1
θ = 90
θ = π/2
Hence the solution is π/2
The equation is an illustration of trigonometry ratios and identities
The solution in the interval [0, 2π) is [tex]\mathbf{\theta = \frac{\pi}{2}}[/tex]
The equation is given as:
[tex]\mathbf{sec\theta - tan \theta = cos\theta}[/tex]
In trigonometry identity, we have:
[tex]\mathbf{sec\theta = \frac{1}{cos\theta}}[/tex]
and
[tex]\mathbf{tan\theta = \frac{sin\theta}{cosθ}}[/tex]
So the equation becomes
[tex]\mathbf{\frac{1}{cos\theta} -\frac{sin\theta}{cos\theta} = cos\theta}[/tex]
Multiply through by cosθ
[tex]\mathbf{1 - sin\theta = cos\²\theta}[/tex]
In trigonometry,
[tex]\mathbf{cos\²\theta = 1 - sin^2\theta }[/tex]
So, we have:
[tex]\mathbf{1-sin\theta = 1-sin\²\theta}[/tex]
Subtract 1 from both sides
[tex]\mathbf{sin\theta = sin\²\theta}[/tex]
Divide both sides by [tex]\mathbf{sin\theta}[/tex]
[tex]\mathbf{1 = sin\theta}[/tex]
Rewrite as:
[tex]\mathbf{sin\theta = 1}[/tex]
Take arc sin of both sides
[tex]\mathbf{\theta = sin^{-1}(1)}[/tex]
This gives
[tex]\mathbf{\theta = \frac{\pi}{2}}[/tex]
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