Solve for x 5cos(theta)=2sec(theta)-3

Problem

5*cos(θ)=2*sec(θ)−3

Solution

1. Substitute the reciprocal identity sec(θ)=1/cos(θ) into the equation.

5*cos(θ)=2⋅1/cos(θ)−3

2. Multiply the entire equation by cos(θ) to eliminate the fraction, assuming cos(θ)≠0

5*cos2(θ)=2−3*cos(θ)

3. Rearrange the terms into a standard quadratic form a*x2+b*x+c=0 by moving all terms to one side.

5*cos2(θ)+3*cos(θ)−2=0

4. Factor the quadratic expression. We look for factors of 5⋅(−2)=−10 that add up to 3 which are 5 and −2

(5*cos(θ)−2)*(cos(θ)+1)=0

5. Set each factor equal to zero to solve for cos(θ)

5*cos(θ)−2=0⇒cos(θ)=2/5

cos(θ)+1=0⇒cos(θ)=−1

6. Solve for θ using the inverse cosine function for each case, where n is any integer.

θ=±arccos(2/5)+2*π*n

θ=π+2*π*n

Final Answer

5*cos(θ)=2*sec(θ)−3⇒θ=±arccos(2/5)+2*π*n,π+2*π*n

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