What is the general solution for cos(A)tan(A)=cos(A) ?

The general solution for cos(A)tan(A)=cos(A) is A= pi/4+pin

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cos(A)tan(A)=cos(A)

cos (A) tan (A) = cos (A)

A = \frac{π}{4} + πn

Degrees

A = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

cos (A) tan (A) = cos (A)

Subtract

cos (A)

from both sides

cos (A) tan (A) - cos (A) = 0

Factor

cos (A) tan (A) - cos (A) : cos (A) (tan (A) - 1)

cos (A) tan (A) - cos (A)

Factor out common term

cos (A)

= cos (A) (tan (A) - 1)

cos (A) (tan (A) - 1) = 0

Solving each part separately

cos (A) = 0 or tan (A) - 1 = 0

cos (A) = 0 : A = \frac{π}{2} + 2 πn, A = \frac{3 π}{2} + 2 πn

cos (A) = 0

General solutions for

cos (A) = 0

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

A = \frac{π}{2} + 2 πn, A = \frac{3 π}{2} + 2 πn

A = \frac{π}{2} + 2 πn, A = \frac{3 π}{2} + 2 πn

tan (A) - 1 = 0 : A = \frac{π}{4} + πn

tan (A) - 1 = 0

Move

1

to the right side

tan (A) - 1 = 0

Add

1

to both sides

tan (A) - 1 + 1 = 0 + 1

Simplify

tan (A) = 1

tan (A) = 1

General solutions for

tan (A) = 1

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

A = \frac{π}{4} + πn

A = \frac{π}{4} + πn

Combine all the solutions

A = \frac{π}{2} + 2 πn, A = \frac{3 π}{2} + 2 πn, A = \frac{π}{4} + πn

Since the equation is undefined for:

\frac{π}{2} + 2 πn, \frac{3 π}{2} + 2 πn

A = \frac{π}{4} + πn

sin (x) = \frac{11 sin ( 12 0 ^{\circ} )}{18}

3 csc (θ) + 5 = csc (θ) + 9

\frac{cos ( 2 x )}{sin ( 3 x ) - sin ( x )} - 1 = 0

2 tan^{3} (x) - \frac{2}{3} tan (x) = 0, 0 \leq x \leq π

6 cos (2 x) - cos (x) - 1 = 0

What is the general solution for cos(A)tan(A)=cos(A) ?
The general solution for cos(A)tan(A)=cos(A) is A= pi/4+pin