What is the general solution for sec(x)+4cos(x)=5 ?

The general solution for sec(x)+4cos(x)=5 is x=1.31811…+2pin,x=2pi-1.31811…+2pin,x=2pin

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sec(x)+4cos(x)=5

sec (x) + 4 cos (x) = 5

x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn, x = 2 πn

Solution steps

sec (x) + 4 cos (x) = 5

Subtract

5

from both sides

sec (x) + 4 cos (x) - 5 = 0

Rewrite using trig identities

- 5 + \frac{4}{sec ( x )} + sec (x) = 0

Solve by substitution

sec (x) = 4, sec (x) = 1

sec (x) = 4 : x = arcsec (4) + 2 πn, x = 2 π - arcsec (4) + 2 πn

sec (x) = 1 : x = 2 πn

Combine all the solutions

x = arcsec (4) + 2 πn, x = 2 π - arcsec (4) + 2 πn, x = 2 πn

Show solutions in decimal form

x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn, x = 2 πn

- 3 sec (x) = 2

3 cos (x) csc (x) = 23 cos (x)

cos (2 x) = \frac{9}{41}

sec (5 x) = \frac{2}{3}

cos (a) = - \frac{4}{9}, \frac{π}{2} \leq a \leq π

What is the general solution for sec(x)+4cos(x)=5 ?
The general solution for sec(x)+4cos(x)=5 is x=1.31811…+2pin,x=2pi-1.31811…+2pin,x=2pin