Find the complete solution of each equation. Express your answer in degrees.
sec^2 θ+sec θ=0
2 cos^2 θ+1=0
cot θ=cot^2 θ
sin^2 θ+5 sin θ=0
sec (sec+1)=0
theta= arcsec0 and theta=arcsec-1
I will be happy to critique your work on the others. Are you certain you copied the second one correctly?
To find the complete solution of each equation, we will solve each equation step by step:
1. sec^2 θ + sec θ = 0:
Rearranging the equation, we get sec^2 θ = -sec θ.
Dividing by sec θ (since sec θ ≠ 0), we have sec θ = -1.
Now, recall that sec θ = 1/cos θ, we can rewrite the equation as 1/cos θ = -1.
Multiplying both sides by cos θ, we get 1 = -cos θ.
Solving for cos θ, we have cos θ = -1.
The solutions for θ that satisfy this equation are θ = 180° + 360°n, where n is an integer.
2. 2 cos^2 θ + 1 = 0:
Subtracting 1 from both sides, we have 2 cos^2 θ = -1.
Dividing by 2, we get cos^2 θ = -1/2.
Since the range of cosine function is [-1, 1], there are no real solutions to this equation.
3. cot θ = cot^2 θ:
Rearranging the equation, we have cot^2 θ - cot θ = 0.
Factoring out cot θ, we get cot θ (cot θ - 1) = 0.
This equation is satisfied when either cot θ = 0 or cot θ - 1 = 0.
Solving cot θ = 0, we find θ = 90° + 180°n, where n is an integer.
Solving cot θ - 1 = 0, we get cot θ = 1, which means θ = 45° + 180°n or θ = -45° + 180°n, where n is an integer.
4. sin^2 θ + 5 sin θ = 0:
Factoring out sin θ, we have sin θ (sin θ + 5) = 0.
This equation is satisfied when either sin θ = 0 or sin θ + 5 = 0.
Solving sin θ = 0, we find θ = 0° + 180°n, where n is an integer.
Solving sin θ + 5 = 0, we get sin θ = -5. Since the range of sine function is [-1, 1], there are no real solutions to this equation.
Therefore, the complete solutions for the given equations in degrees are:
1. θ = 180° + 360°n, where n is an integer.
2. No real solutions.
3. θ = 90° + 180°n, θ = 45° + 180°n, or θ = -45° + 180°n, where n is an integer.
4. θ = 0° + 180°n, where n is an integer.