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Popular >

sec^2(x)-4tan^2(x)=0

Solution

sec^{2} (x) - 4 tan^{2} (x) = 0

Solution

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Degrees

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

sec^{2} (x) - 4 tan^{2} (x) = 0

Factor

sec^{2} (x) - 4 tan^{2} (x) : (sec (x) + 2 tan (x)) (sec (x) - 2 tan (x))

sec^{2} (x) - 4 tan^{2} (x)

Rewrite

sec^{2} (x) - 4 tan^{2} (x)

sec^{2} (x) - (2 tan (x))^{2}

sec^{2} (x) - 4 tan^{2} (x)

Rewrite

4

2^{2}

= sec^{2} (x) - 2^{2} tan^{2} (x)

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

2^{2} tan^{2} (x) = (2 tan (x))^{2}

= sec^{2} (x) - (2 tan (x))^{2}

= sec^{2} (x) - (2 tan (x))^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

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

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

Solving each part separately

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

sec (x) + 2 tan (x) = 0 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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

Express with sin, cos

sec (x) + 2 tan (x)

Use the basic trigonometric identity:

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

= \frac{1}{cos ( x )} + 2 tan (x)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{1}{cos ( x )} + 2 \cdot \frac{sin ( x )}{cos ( x )}

Simplify

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

\frac{1}{cos ( x )} + 2 \cdot \frac{sin ( x )}{cos ( x )}

Multiply

2 \cdot \frac{sin ( x )}{cos ( x )} : \frac{2 sin ( x )}{cos ( x )}

2 \cdot \frac{sin ( x )}{cos ( x )}

Multiply fractions:

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

= \frac{sin ( x ) \cdot 2}{cos ( x )}

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

Apply rule

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

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

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

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

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

1 + 2 sin (x) = 0

Move

1

to the right side

1 + 2 sin (x) = 0

Subtract

1

from both sides

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

Simplify

2 sin (x) = - 1

2 sin (x) = - 1

Divide both sides by

2

2 sin (x) = - 1

Divide both sides by

2

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

Simplify

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

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

General solutions for

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

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sec (x) - 2 tan (x) = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

Express with sin, cos

sec (x) - 2 tan (x)

Use the basic trigonometric identity:

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

= \frac{1}{cos ( x )} - 2 tan (x)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{1}{cos ( x )} - 2 \cdot \frac{sin ( x )}{cos ( x )}

Simplify

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

\frac{1}{cos ( x )} - 2 \cdot \frac{sin ( x )}{cos ( x )}

Multiply

2 \cdot \frac{sin ( x )}{cos ( x )} : \frac{2 sin ( x )}{cos ( x )}

2 \cdot \frac{sin ( x )}{cos ( x )}

Multiply fractions:

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

= \frac{sin ( x ) \cdot 2}{cos ( x )}

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

Apply rule

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

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

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

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

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

1 - 2 sin (x) = 0

Move

1

to the right side

1 - 2 sin (x) = 0

Subtract

1

from both sides

1 - 2 sin (x) - 1 = 0 - 1

Simplify

- 2 sin (x) = - 1

- 2 sin (x) = - 1

Divide both sides by

- 2

- 2 sin (x) = - 1

Divide both sides by

- 2

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

Simplify

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \frac{1}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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