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Popular >

sec^2(x)-2tan^2(x)= 2/3

Solution

sec^{2} (x) - 2 tan^{2} (x) = \frac{2}{3}

Solution

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

Degrees

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

Solution steps

sec^{2} (x) - 2 tan^{2} (x) = \frac{2}{3}

Subtract

\frac{2}{3}

from both sides

sec^{2} (x) - 2 tan^{2} (x) - \frac{2}{3} = 0

Simplify

sec^{2} (x) - 2 tan^{2} (x) - \frac{2}{3} : \frac{3 sec ^{2} ( x ) - 6 tan ^{2} ( x ) - 2}{3}

sec^{2} (x) - 2 tan^{2} (x) - \frac{2}{3}

Convert element to fraction:

sec^{2} (x) = \frac{sec ^{2} ( x ) 3}{3}, 2 tan^{2} (x) = \frac{2 tan ^{2} ( x ) 3}{3}

= \frac{sec ^{2} ( x ) \cdot 3}{3} - \frac{2 tan ^{2} ( x ) \cdot 3}{3} - \frac{2}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{sec ^{2} ( x ) \cdot 3 - 2 tan ^{2} ( x ) \cdot 3 - 2}{3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{3 sec ^{2} ( x ) - 6 tan ^{2} ( x ) - 2}{3}

\frac{3 sec ^{2} ( x ) - 6 tan ^{2} ( x ) - 2}{3} = 0

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

3 sec^{2} (x) - 6 tan^{2} (x) - 2 = 0

Rewrite using trig identities

- 2 + 3 sec^{2} (x) - 6 tan^{2} (x)

Use the Pythagorean identity:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

= - 2 + 3 sec^{2} (x) - 6 (sec^{2} (x) - 1)

Simplify

- 2 + 3 sec^{2} (x) - 6 (sec^{2} (x) - 1) : - 3 sec^{2} (x) + 4

- 2 + 3 sec^{2} (x) - 6 (sec^{2} (x) - 1)

Expand

- 6 (sec^{2} (x) - 1) : - 6 sec^{2} (x) + 6

- 6 (sec^{2} (x) - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = - 6, b = sec^{2} (x), c = 1

= - 6 sec^{2} (x) - (- 6) \cdot 1

Apply minus-plus rules

- (- a) = a

= - 6 sec^{2} (x) + 6 \cdot 1

Multiply the numbers:

6 \cdot 1 = 6

= - 6 sec^{2} (x) + 6

= - 2 + 3 sec^{2} (x) - 6 sec^{2} (x) + 6

Simplify

- 2 + 3 sec^{2} (x) - 6 sec^{2} (x) + 6 : - 3 sec^{2} (x) + 4

- 2 + 3 sec^{2} (x) - 6 sec^{2} (x) + 6

Add similar elements:

3 sec^{2} (x) - 6 sec^{2} (x) = - 3 sec^{2} (x)

= - 2 - 3 sec^{2} (x) + 6

Group like terms

= - 3 sec^{2} (x) - 2 + 6

Add/Subtract the numbers:

- 2 + 6 = 4

= - 3 sec^{2} (x) + 4

= - 3 sec^{2} (x) + 4

= - 3 sec^{2} (x) + 4

4 - 3 sec^{2} (x) = 0

Solve by substitution

4 - 3 sec^{2} (x) = 0

Let:

sec (x) = u

4 - 3 u^{2} = 0

4 - 3 u^{2} = 0 : u = \frac{2 3}{3}, u = - \frac{2 3}{3}

4 - 3 u^{2} = 0

Move

4

to the right side

4 - 3 u^{2} = 0

Subtract

4

from both sides

4 - 3 u^{2} - 4 = 0 - 4

Simplify

- 3 u^{2} = - 4

- 3 u^{2} = - 4

Divide both sides by

- 3

- 3 u^{2} = - 4

Divide both sides by

- 3

\frac{- 3 u ^{2}}{- 3} = \frac{- 4}{- 3}

Simplify

u^{2} = \frac{4}{3}

u^{2} = \frac{4}{3}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{4}{3}, u = - \frac{4}{3}

\frac{4}{3} = \frac{2 3}{3}

\frac{4}{3}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{4}{3}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2}{3}

Rationalize

\frac{2}{3} : \frac{2 3}{3}

\frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{2 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{2 3}{3}

= \frac{2 3}{3}

- \frac{4}{3} = - \frac{2 3}{3}

- \frac{4}{3}

Simplify

\frac{4}{3} : \frac{2}{3}

\frac{4}{3}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{4}{3}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2}{3}

= - \frac{2}{3}

Rationalize

- \frac{2}{3} : - \frac{2 3}{3}

- \frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{2 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{2 3}{3}

= - \frac{2 3}{3}

u = \frac{2 3}{3}, u = - \frac{2 3}{3}

Substitute back

u = sec (x)

sec (x) = \frac{2 3}{3}, sec (x) = - \frac{2 3}{3}

sec (x) = \frac{2 3}{3}, sec (x) = - \frac{2 3}{3}

sec (x) = \frac{2 3}{3} : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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

General solutions for

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

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

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

sec (x) = - \frac{2 3}{3} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

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

General solutions for

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

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

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

Combine all the solutions

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

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