What is the general solution for -sec(x)+sqrt(2)sec(x)cos(x)=0 ?

The general solution for -sec(x)+sqrt(2)sec(x)cos(x)=0 is x= pi/4+2pin,x=(7pi)/4+2pin

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-sec(x)+sqrt(2)sec(x)cos(x)=0

- sec (x) + 2 sec (x) cos (x) = 0

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

- sec (x) + 2 sec (x) cos (x) = 0

Rewrite using trig identities

- sec (x) + 2 sec (x) cos (x)

sec (x) cos (x) = 1

sec (x) cos (x)

Express with sin, cos

sec (x) cos (x)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= \frac{1}{cos ( x )} cos (x)

\frac{1}{cos ( x )} cos (x) = 1

\frac{1}{cos ( x )} cos (x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 cos ( x )}{cos ( x )}

Cancel the common factor:

cos (x)

= 1

= 1

= - sec (x) + 2 \cdot 1

- sec (x) + 2 \cdot 1 = 0

Multiply:

2 \cdot 1 = 2

- sec (x) + 2 = 0

Move

2

to the right side

- sec (x) + 2 = 0

Subtract

2

from both sides

- sec (x) + 2 - 2 = 0 - 2

Simplify

- sec (x) = - 2

- sec (x) = - 2

Divide both sides by

- 1

- sec (x) = - 2

Divide both sides by

- 1

\frac{- sec ( x )}{- 1} = \frac{- 2}{- 1}

Simplify

sec (x) = 2

sec (x) = 2

General solutions for

sec (x) = 2

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

2 tan (x) - 3 sin (x) = 0

sin (x) + 5 = 4

- 9 cos (x) = 0

- 3 cos (x) = 0

4 cot (x) + 2 = 6

What is the general solution for -sec(x)+sqrt(2)sec(x)cos(x)=0 ?
The general solution for -sec(x)+sqrt(2)sec(x)cos(x)=0 is x= pi/4+2pin,x=(7pi)/4+2pin