This is an equation in trigonometry that involves the cosine and sine functions of an angle θ. The equation is cos θ + 2 sin θ = 2. Solving for θ requires knowledge of trigonometric identities and algebraic manipulation. If you have any questions about how to solve this equation, feel free to ask!
How do I solve cos θ + 2 sin θ = 2?
To solve the equation cos θ + 2 sin θ = 2, we can use the fact that sin^2 θ + cos^2 θ = 1.
First, we can square both sides of the equation to get rid of the square root:
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(cos θ + 2 sin θ)^2 = 2^2
Expanding the left side of the equation gives:
cos^2 θ + 4 sin θ cos θ + 4 sin^2 θ = 4
Next, we can use the identity sin^2 θ + cos^2 θ = 1 to substitute for cos^2 θ:
1 – sin^2 θ + 4 sin θ cos θ + 4 sin^2 θ = 4
Simplifying the equation gives:
5 sin^2 θ + 4 sin θ cos θ – 3 = 0
We can then use the quadratic formula to solve for sin θ:
sin θ = (-b ± sqrt(b^2 – 4ac)) / 2a
where a = 5, b = 4 cos θ, and c = -3.
Substituting these values into the formula and simplifying gives:
sin θ = (-2 ± sqrt(16 cos^2 θ + 60)) / 10
We can then solve for cos θ using the identity cos^2 θ + sin^2 θ = 1:
cos θ = ± sqrt(1 – sin^2 θ)
Substituting the expression for sin θ into this formula gives:
cos θ = ± sqrt(1 – (-2 ± sqrt(16 cos^2 θ + 60)) / 10)^2
This is a bit messy, but we can simplify it by making the substitution u = cos^2 θ:
cos θ = ± sqrt(1 – (-2 ± 2 sqrt(4u + 15)) / 10)^2
cos θ = ± sqrt(1 + (-2 ± 2 sqrt(4u + 15)) / 10)^2
cos θ = ± (1 + (-2 ± 2 sqrt(4u + 15)) / 10)
We can then use the fact that cos^2 θ + sin^2 θ = 1 to solve for sin θ:
sin θ = ± sqrt(1 – cos^2 θ)
Substituting the expression for cos θ into this formula gives:
sin θ = ± sqrt(1 – (1 + (-2 ± 2 sqrt(4u + 15)) / 10)^2)
This gives us the solutions for sin θ and cos θ in terms of u. We can then solve for u using the original equation:
cos θ + 2 sin θ = 2
Substituting the expressions for sin θ and cos θ in terms of u gives:
sqrt(u) ± 2 sqrt(1 – u) = 2
Squaring both sides and simplifying gives:
5u^2 – 8u + 3 = 0
This is a quadratic equation that can be factored as:
(5u – 3)(u – 1) = 0
Therefore, the solutions for u are u = 3/5 and u = 1.
Substituting these values back into the expressions for sin θ and cos θ gives:
sin θ = ± sqrt(1 – cos^2 θ) = ± sqrt(16/25) = ± 4/5
cos θ = ± sqrt(1 – sin^2 θ) = ± sqrt(9/25) = ± 3/5
Therefore, the solutions