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Popular >

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

Solution

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

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

+1

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

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

Subtract

3

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Add similar elements:

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

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

Group like terms

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

Subtract the numbers:

- 3 - 1 = - 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

Add

4

to 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

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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