What is the general solution for tan(2x)+sec(2x)=11 ?

The general solution for tan(2x)+sec(2x)=11 is x=(1.38947…)/2+pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

tan(2x)+sec(2x)=11

tan (2 x) + sec (2 x) = 11

x = \frac{1.38947 \dots}{2} + πn

Solution steps

tan (2 x) + sec (2 x) = 11

Subtract

11

from both sides

tan (2 x) + sec (2 x) - 11 = 0

Express with sin, cos

\frac{sin ( 2 x )}{cos ( 2 x )} + \frac{1}{cos ( 2 x )} - 11 = 0

Simplify

\frac{sin ( 2 x )}{cos ( 2 x )} + \frac{1}{cos ( 2 x )} - 11 : \frac{sin ( 2 x ) + 1 - 11 cos ( 2 x )}{cos ( 2 x )}

\frac{sin ( 2 x ) + 1 - 11 cos ( 2 x )}{cos ( 2 x )} = 0

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

sin (2 x) + 1 - 11 cos (2 x) = 0

Add

11 cos (2 x)

to both sides

sin (2 x) + 1 = 11 cos (2 x)

Square both sides

(sin (2 x) + 1)^{2} = (11 cos (2 x))^{2}

Subtract

(11 cos (2 x))^{2}

from both sides

(sin (2 x) + 1)^{2} - 121 cos^{2} (2 x) = 0

Rewrite using trig identities

- 120 + 122 sin^{2} (2 x) + 2 sin (2 x) = 0

Solve by substitution

sin (2 x) = \frac{60}{61}, sin (2 x) = - 1

sin (2 x) = \frac{60}{61} : x = \frac{arcsin ( \frac{60}{61} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{60}{61} )}{2} + πn

sin (2 x) = - 1 : x = \frac{3 π}{4} + πn

Combine all the solutions

x = \frac{arcsin ( \frac{60}{61} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{60}{61} )}{2} + πn, x = \frac{3 π}{4} + πn

Verify solutions by plugging them into the original equation

x = \frac{arcsin ( \frac{60}{61} )}{2} + πn

Show solutions in decimal form

x = \frac{1.38947 \dots}{2} + πn

- 8 sin (x) + 8 cos (2 x) = 0

tan^{2} (x) + 2 sec (x) + 2 = 0

sin (θ) = 1 + cos (θ)

7 cos (2 x) = 7 cos^{2} (x) - 3

- cos^{2} (θ) + 1 = 0

What is the general solution for tan(2x)+sec(2x)=11 ?
The general solution for tan(2x)+sec(2x)=11 is x=(1.38947…)/2+pin