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Popular >

sec^2(x)+tan^2(x)=7

Solution

sec^{2} (x) + tan^{2} (x) = 7

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

+1

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) + tan^{2} (x) = 7

Subtract

7

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

- 7 + sec^{2} (x) + sec^{2} (x) - 1 : 2 sec^{2} (x) - 8

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

Add similar elements:

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

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

Group like terms

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

Subtract the numbers:

- 7 - 1 = - 8

= 2 sec^{2} (x) - 8

= 2 sec^{2} (x) - 8

- 8 + 2 sec^{2} (x) = 0

Solve by substitution

- 8 + 2 sec^{2} (x) = 0

Let:

sec (x) = u

- 8 + 2 u^{2} = 0

- 8 + 2 u^{2} = 0 : u = 2, u = - 2

- 8 + 2 u^{2} = 0

Move

8

to the right side

- 8 + 2 u^{2} = 0

Add

8

to both sides

- 8 + 2 u^{2} + 8 = 0 + 8

Simplify

2 u^{2} = 8

2 u^{2} = 8

Divide both sides by

2

2 u^{2} = 8

Divide both sides by

2

\frac{2 u ^{2}}{2} = \frac{8}{2}

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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