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Popular >

sec^2(θ)+tan^2(θ)=(2)^2

Solution

sec^{2} (θ) + tan^{2} (θ) = (2)^{2}

Solution

θ = 0.88607 \dots + 2 πn, θ = 2 π - 0.88607 \dots + 2 πn, θ = 2.25551 \dots + 2 πn, θ = - 2.25551 \dots + 2 πn

+1

Degrees

θ = 50.76847 \dots^{\circ} + 36 0^{\circ} n, θ = 309.23152 \dots^{\circ} + 36 0^{\circ} n, θ = 129.23152 \dots^{\circ} + 36 0^{\circ} n, θ = - 129.23152 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sec^{2} (θ) + tan^{2} (θ) = (2)^{2}

Subtract

2^{2}

from both sides

sec^{2} (θ) + tan^{2} (θ) - 4 = 0

Rewrite using trig identities

- 4 + sec^{2} (θ) + tan^{2} (θ)

Use the Pythagorean identity:

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

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

= - 4 + sec^{2} (θ) + sec^{2} (θ) - 1

Simplify

- 4 + sec^{2} (θ) + sec^{2} (θ) - 1 : 2 sec^{2} (θ) - 5

- 4 + sec^{2} (θ) + sec^{2} (θ) - 1

Add similar elements:

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

= - 4 + 2 sec^{2} (θ) - 1

Group like terms

= 2 sec^{2} (θ) - 4 - 1

Subtract the numbers:

- 4 - 1 = - 5

= 2 sec^{2} (θ) - 5

= 2 sec^{2} (θ) - 5

- 5 + 2 sec^{2} (θ) = 0

Solve by substitution

- 5 + 2 sec^{2} (θ) = 0

Let:

sec (θ) = u

- 5 + 2 u^{2} = 0

- 5 + 2 u^{2} = 0 : u = \frac{5}{2}, u = - \frac{5}{2}

- 5 + 2 u^{2} = 0

Move

5

to the right side

- 5 + 2 u^{2} = 0

Add

5

to both sides

- 5 + 2 u^{2} + 5 = 0 + 5

Simplify

2 u^{2} = 5

2 u^{2} = 5

Divide both sides by

2

2 u^{2} = 5

Divide both sides by

2

\frac{2 u ^{2}}{2} = \frac{5}{2}

Simplify

u^{2} = \frac{5}{2}

u^{2} = \frac{5}{2}

For

x^{2} = f (a)

the solutions are

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

u = \frac{5}{2}, u = - \frac{5}{2}

Substitute back

u = sec (θ)

sec (θ) = \frac{5}{2}, sec (θ) = - \frac{5}{2}

sec (θ) = \frac{5}{2}, sec (θ) = - \frac{5}{2}

sec (θ) = \frac{5}{2} : θ = arcsec (\frac{5}{2}) + 2 πn, θ = 2 π - arcsec (\frac{5}{2}) + 2 πn

sec (θ) = \frac{5}{2}

Apply trig inverse properties

sec (θ) = \frac{5}{2}

General solutions for

sec (θ) = \frac{5}{2}

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = 2 π - arcsec (a) + 2 πn

θ = arcsec (\frac{5}{2}) + 2 πn, θ = 2 π - arcsec (\frac{5}{2}) + 2 πn

θ = arcsec (\frac{5}{2}) + 2 πn, θ = 2 π - arcsec (\frac{5}{2}) + 2 πn

sec (θ) = - \frac{5}{2} : θ = arcsec (- \frac{5}{2}) + 2 πn, θ = - arcsec (- \frac{5}{2}) + 2 πn

sec (θ) = - \frac{5}{2}

Apply trig inverse properties

sec (θ) = - \frac{5}{2}

General solutions for

sec (θ) = - \frac{5}{2}

sec (x) = - a \Rightarrow x = arcsec (- a) + 2 πn, x = - arcsec (- a) + 2 πn

θ = arcsec (- \frac{5}{2}) + 2 πn, θ = - arcsec (- \frac{5}{2}) + 2 πn

θ = arcsec (- \frac{5}{2}) + 2 πn, θ = - arcsec (- \frac{5}{2}) + 2 πn

Combine all the solutions

θ = arcsec (\frac{5}{2}) + 2 πn, θ = 2 π - arcsec (\frac{5}{2}) + 2 πn, θ = arcsec (- \frac{5}{2}) + 2 πn, θ = - arcsec (- \frac{5}{2}) + 2 πn

Show solutions in decimal form

θ = 0.88607 \dots + 2 πn, θ = 2 π - 0.88607 \dots + 2 πn, θ = 2.25551 \dots + 2 πn, θ = - 2.25551 \dots + 2 πn

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