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Popular >

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

Solution

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

Solution

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

+1

Degrees

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

Solution steps

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

Subtract

2 sec^{2} (y)

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

sec^{2} (y) - 2 sec^{2} (y) = - sec^{2} (y)

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

Add/Subtract the numbers:

3 - 1 = 2

= - sec^{2} (y) + 2

= - sec^{2} (y) + 2

2 - sec^{2} (y) = 0

Solve by substitution

2 - sec^{2} (y) = 0

Let:

sec (y) = u

2 - u^{2} = 0

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

2 - u^{2} = 0

Move

2

to the right side

2 - u^{2} = 0

Subtract

2

from both sides

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

Simplify

- u^{2} = - 2

- u^{2} = - 2

Divide both sides by

- 1

- u^{2} = - 2

Divide both sides by

- 1

\frac{- u ^{2}}{- 1} = \frac{- 2}{- 1}

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 (y)

sec (y) = 2, sec (y) = - 2

sec (y) = 2, sec (y) = - 2

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

sec (y) = 2

General solutions for

sec (y) = 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}

y = \frac{π}{4} + 2 πn, y = \frac{7 π}{4} + 2 πn

y = \frac{π}{4} + 2 πn, y = \frac{7 π}{4} + 2 πn

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

sec (y) = - 2

General solutions for

sec (y) = - 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}

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

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

Combine all the solutions

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

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