What is the general solution for sec(x)=3cos(x)-tan(x) ?

The general solution for sec(x)=3cos(x)-tan(x) is x=0.72972…+2pin,x=pi-0.72972…+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sec(x)=3cos(x)-tan(x)

sec (x) = 3 cos (x) - tan (x)

x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

Solution steps

sec (x) = 3 cos (x) - tan (x)

Subtract

3 cos (x) - tan (x)

from both sides

sec (x) - 3 cos (x) + tan (x) = 0

Express with sin, cos

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

Simplify

\frac{1}{cos ( x )} - 3 cos (x) + \frac{sin ( x )}{cos ( x )} : \frac{1 + sin ( x ) - 3 cos ^{2} ( x )}{cos ( x )}

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

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

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

Add

3 cos^{2} (x)

to both sides

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

Square both sides

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

Subtract

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

from both sides

(1 + sin (x))^{2} - 9 cos^{4} (x) = 0

Factor

(1 + sin (x))^{2} - 9 cos^{4} (x) : (1 + sin (x) + 3 cos^{2} (x)) (1 + sin (x) - 3 cos^{2} (x))

(1 + sin (x) + 3 cos^{2} (x)) (1 + sin (x) - 3 cos^{2} (x)) = 0

Solving each part separately

1 + sin (x) + 3 cos^{2} (x) = 0 or 1 + sin (x) - 3 cos^{2} (x) = 0

1 + sin (x) + 3 cos^{2} (x) = 0 : x = \frac{3 π}{2} + 2 πn

1 + sin (x) - 3 cos^{2} (x) = 0 : x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn, x = \frac{3 π}{2} + 2 πn

Combine all the solutions

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

Verify solutions by plugging them into the original equation

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

Since the equation is undefined for:

\frac{3 π}{2} + 2 πn

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

Show solutions in decimal form

x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

arctan (x^{2} - 2 x) = 0

sin (x^{\circ}) = \frac{6.57 \cdot 1 0 ^{- 7}}{1.8 \cdot 1 0 ^{- 6}}

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

y, x tan (y) - x^{2} sin (x) + 3 x^{3} = 0

(\frac{( 1 - sin ^{2} ( x ) )}{( 1 + 2 \cdot sin ^{2} ( x ) )}) = 0.45

What is the general solution for sec(x)=3cos(x)-tan(x) ?
The general solution for sec(x)=3cos(x)-tan(x) is x=0.72972…+2pin,x=pi-0.72972…+2pin