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Popular >

sec^2(x)+3tan^2(x)=13

Solution

sec^{2} (x) + 3 tan^{2} (x) = 13

Solution

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

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

Solution steps

sec^{2} (x) + 3 tan^{2} (x) = 13

Subtract

13

from both sides

sec^{2} (x) + 3 tan^{2} (x) - 13 = 0

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

= - 13 + sec^{2} (x) + 3 (sec^{2} (x) - 1)

Simplify

- 13 + sec^{2} (x) + 3 (sec^{2} (x) - 1) : 4 sec^{2} (x) - 16

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

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

= - 13 + sec^{2} (x) + 3 sec^{2} (x) - 3

Simplify

- 13 + sec^{2} (x) + 3 sec^{2} (x) - 3 : 4 sec^{2} (x) - 16

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

Add similar elements:

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

= - 13 + 4 sec^{2} (x) - 3

Group like terms

= 4 sec^{2} (x) - 13 - 3

Subtract the numbers:

- 13 - 3 = - 16

= 4 sec^{2} (x) - 16

= 4 sec^{2} (x) - 16

= 4 sec^{2} (x) - 16

- 16 + 4 sec^{2} (x) = 0

Solve by substitution

- 16 + 4 sec^{2} (x) = 0

Let:

sec (x) = u

- 16 + 4 u^{2} = 0

- 16 + 4 u^{2} = 0 : u = 2, u = - 2

- 16 + 4 u^{2} = 0

Move

16

to the right side

- 16 + 4 u^{2} = 0

Add

16

to both sides

- 16 + 4 u^{2} + 16 = 0 + 16

Simplify

4 u^{2} = 16

4 u^{2} = 16

Divide both sides by

4

4 u^{2} = 16

Divide both sides by

4

\frac{4 u ^{2}}{4} = \frac{16}{4}

Simplify

u^{2} = 4

u^{2} = 4

For

x^{2} = f (a)

the solutions are

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

u = 4, u = - 4

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

- 4 = - 2

- 4

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= - 2

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

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

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

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

Combine all the solutions

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

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