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Popular >

3tan^2(x)-2=5sec^2(x)-9

Solution

3 tan^{2} (x) - 2 = 5 sec^{2} (x) - 9

Solution

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n

Solution steps

3 tan^{2} (x) - 2 = 5 sec^{2} (x) - 9

Subtract

5 sec^{2} (x) - 9

from both sides

3 tan^{2} (x) - 5 sec^{2} (x) + 7 = 0

Rewrite using trig identities

7 + 3 tan^{2} (x) - 5 sec^{2} (x)

Use the Pythagorean identity:

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

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

= 7 + 3 (sec^{2} (x) - 1) - 5 sec^{2} (x)

Simplify

7 + 3 (sec^{2} (x) - 1) - 5 sec^{2} (x) : - 2 sec^{2} (x) + 4

7 + 3 (sec^{2} (x) - 1) - 5 sec^{2} (x)

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = sec^{2} (x), c = 1

= 3 sec^{2} (x) - 3 \cdot 1

Multiply the numbers:

3 \cdot 1 = 3

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

= 7 + 3 sec^{2} (x) - 3 - 5 sec^{2} (x)

Simplify

7 + 3 sec^{2} (x) - 3 - 5 sec^{2} (x) : - 2 sec^{2} (x) + 4

7 + 3 sec^{2} (x) - 3 - 5 sec^{2} (x)

Group like terms

= 3 sec^{2} (x) - 5 sec^{2} (x) + 7 - 3

Add similar elements:

3 sec^{2} (x) - 5 sec^{2} (x) = - 2 sec^{2} (x)

= - 2 sec^{2} (x) + 7 - 3

Add/Subtract the numbers:

7 - 3 = 4

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

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

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

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

Solve by substitution

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

Let:

sec (x) = u

4 - 2 u^{2} = 0

4 - 2 u^{2} = 0 : u = 2, u = - 2

4 - 2 u^{2} = 0

Move

4

to the right side

4 - 2 u^{2} = 0

Subtract

4

from both sides

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

Simplify

- 2 u^{2} = - 4

- 2 u^{2} = - 4

Divide both sides by

- 2

- 2 u^{2} = - 4

Divide both sides by

- 2

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

Simplify

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

Simplify

\frac{- 2 u ^{2}}{- 2} : u^{2}

\frac{- 2 u ^{2}}{- 2}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

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

Divide the numbers:

\frac{2}{2} = 1

= u^{2}

Simplify

\frac{- 4}{- 2} : 2

\frac{- 4}{- 2}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

u^{2} = 2

u^{2} = 2

u^{2} = 2

For

x^{2} = f (a)

the solutions are

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

u = 2, u = - 2

Substitute back

u = sec (x)

sec (x) = 2, sec (x) = - 2

sec (x) = 2, sec (x) = - 2

sec (x) = 2 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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

sec (x) = - 2 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

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{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Combine all the solutions

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

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