What is the general solution for ((3))/((cos(x)))=8-5tan(x) ?

The general solution for ((3))/((cos(x)))=8-5tan(x) is x=0.68857…+2pin,x=pi+1.33581…+2pin

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((3))/((cos(x)))=8-5tan(x)

\frac{( 3 )}{( cos ( x ) )} = 8 - 5 tan (x)

x = 0.68857 \dots + 2 πn, x = π + 1.33581 \dots + 2 πn

Solution steps

\frac{( 3 )}{( cos ( x ) )} = 8 - 5 tan (x)

Subtract

8 - 5 tan (x)

from both sides

\frac{3}{cos ( x )} - 8 + 5 tan (x) = 0

Simplify

\frac{3}{cos ( x )} - 8 + 5 tan (x) : \frac{3 - 8 cos ( x ) + 5 tan ( x ) cos ( x )}{cos ( x )}

\frac{3 - 8 cos ( x ) + 5 tan ( x ) cos ( x )}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 - 8 cos (x) + 5 tan (x) cos (x) = 0

Express with sin, cos

3 - 8 cos (x) + 5 \cdot \frac{sin ( x )}{cos ( x )} cos (x) = 0

Simplify

3 - 8 cos (x) + 5 \cdot \frac{sin ( x )}{cos ( x )} cos (x) : 3 - 8 cos (x) + 5 sin (x)

3 - 8 cos (x) + 5 sin (x) = 0

Add

8 cos (x)

to both sides

5 sin (x) + 3 = 8 cos (x)

Square both sides

(5 sin (x) + 3)^{2} = (8 cos (x))^{2}

Subtract

(8 cos (x))^{2}

from both sides

(5 sin (x) + 3)^{2} - 64 cos^{2} (x) = 0

Rewrite using trig identities

- 55 + 30 sin (x) + 89 sin^{2} (x) = 0

Solve by substitution

sin (x) = \frac{- 15 + 32 5}{89}, sin (x) = - \frac{15 + 32 5}{89}

sin (x) = \frac{- 15 + 32 5}{89} : x = arcsin (\frac{- 15 + 32 5}{89}) + 2 πn, x = π - arcsin (\frac{- 15 + 32 5}{89}) + 2 πn

sin (x) = - \frac{15 + 32 5}{89} : x = arcsin (- \frac{15 + 32 5}{89}) + 2 πn, x = π + arcsin (\frac{15 + 32 5}{89}) + 2 πn

Combine all the solutions

x = arcsin (\frac{- 15 + 32 5}{89}) + 2 πn, x = π - arcsin (\frac{- 15 + 32 5}{89}) + 2 πn, x = arcsin (- \frac{15 + 32 5}{89}) + 2 πn, x = π + arcsin (\frac{15 + 32 5}{89}) + 2 πn

Verify solutions by plugging them into the original equation

x = arcsin (\frac{- 15 + 32 5}{89}) + 2 πn, x = π + arcsin (\frac{15 + 32 5}{89}) + 2 πn

Show solutions in decimal form

x = 0.68857 \dots + 2 πn, x = π + 1.33581 \dots + 2 πn

x, π + 3 arccos (x + 1) = 0

cos (x + 4 0^{\circ}) = 0.3

cos (3 x) = - 1, 0 \leq x \leq 2 π

sin (x) = - 3 cos (x)

4 sin^{2} (x) + 2 cos^{2} (x) = 3, 0 \leq x \leq 2 π

What is the general solution for ((3))/((cos(x)))=8-5tan(x) ?
The general solution for ((3))/((cos(x)))=8-5tan(x) is x=0.68857…+2pin,x=pi+1.33581…+2pin