Solve Trig. Problem without a calculator

[morr485]

1. sec^2(x) = (1 + sqrt3) - (1 - sqrt3)*tan (x)



2.sec^2(x) = tan^2(x) +1



3. tan^2(x) + tan(x) + (1 - sqrt3)*tan(x) - sqrt3 = 0
tan^(x)[tan(x) +1 +1 - sqrt3) -sqrt(3) = 0
tan^2(x) + (2 - sqrt3) - sqrt(3) = 0

 

[Mark44]

1. sec^2(x) = (1 + sqrt3) - (1 - sqrt3)*tan (x)



2.sec^2(x) = tan^2(x) +1



3. tan^2(x) + tan(x) + (1 - sqrt3)*tan(x) - sqrt3 = 0
tan^(x)[tan(x) +1 +1 - sqrt3) -sqrt(3) = 0
tan^2(x) + (2 - sqrt3) - sqrt(3) = 0

sec²(x) - 1 + (1 - sqrt(3))tan(x) - sqrt(3) = 0

Replace sec²(x) - 1 with tan²(x) and you will have a quadratic equation in tan(x), which you can solve by factoring (maybe) or by use of the quadratic formula.

 

[The Tutor]

For #2, sec^2 will equal 1/cos^2, and tan^2 will equal sin^2/cos^2. If you have learned the Pythagorean Identity (sin^2+cos^2=1) you can isolate cos^2 and solve the left side, which will leave you with 1/1-sin^2. Solving the right side, tan^2 will equal sin^2/cos^2, and you can make 1 equal to sin^2/cos^2, which would equal 2sin^2/cos^2. You can solve this equation to prove the left side.