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Popular Trigonometry >

1-tan^2(y)=2sec(y)-4

Solution

1 - tan^{2} (y) = 2 sec (y) - 4

Solution

y = 1.84864 \dots + 2 πn, y = - 1.84864 \dots + 2 πn, y = 0.91772 \dots + 2 πn, y = 2 π - 0.91772 \dots + 2 πn

Degrees

y = 105.91981 \dots^{\circ} + 36 0^{\circ} n, y = - 105.91981 \dots^{\circ} + 36 0^{\circ} n, y = 52.58201 \dots^{\circ} + 36 0^{\circ} n, y = 307.41798 \dots^{\circ} + 36 0^{\circ} n

Solution steps

1 - tan^{2} (y) = 2 sec (y) - 4

Subtract

2 sec (y) - 4

from both sides

- tan^{2} (y) - 2 sec (y) + 5 = 0

Rewrite using trig identities

5 - tan^{2} (y) - 2 sec (y)

Use the Pythagorean identity:

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

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

= 5 - (sec^{2} (y) - 1) - 2 sec (y)

Simplify

5 - (sec^{2} (y) - 1) - 2 sec (y) : - sec^{2} (y) - 2 sec (y) + 6

5 - (sec^{2} (y) - 1) - 2 sec (y)

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

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

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

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

= 5 - sec^{2} (y) + 1 - 2 sec (y)

Simplify

5 - sec^{2} (y) + 1 - 2 sec (y) : - sec^{2} (y) - 2 sec (y) + 6

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

Group like terms

= - sec^{2} (y) - 2 sec (y) + 5 + 1

Add the numbers:

5 + 1 = 6

= - sec^{2} (y) - 2 sec (y) + 6

= - sec^{2} (y) - 2 sec (y) + 6

= - sec^{2} (y) - 2 sec (y) + 6

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

Solve by substitution

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

Let:

sec (y) = u

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

6 - u^{2} - 2 u = 0 : u = - 1 - 7, u = 7 - 1

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

Write in the standard form

a x^{2} + b x + c = 0

- u^{2} - 2 u + 6 = 0

Solve with the quadratic formula

- u^{2} - 2 u + 6 = 0

Quadratic Equation Formula:

For

a = - 1, b = - 2, c = 6

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 ( - 1 ) \cdot 6}{2 ( - 1 )}

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 ( - 1 ) \cdot 6}{2 ( - 1 )}

(- 2)^{2} - 4 (- 1) \cdot 6 = 27

(- 2)^{2} - 4 (- 1) \cdot 6

Apply rule

- (- a) = a

= (- 2)^{2} + 4 \cdot 1 \cdot 6

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

= 2^{2} + 4 \cdot 1 \cdot 6

Multiply the numbers:

4 \cdot 1 \cdot 6 = 24

= 2^{2} + 24

2^{2} = 4

= 4 + 24

Add the numbers:

4 + 24 = 28

= 28

Prime factorization of

28 : 2^{2} \cdot 7

28

28

divides by

228 = 14 \cdot 2

= 2 \cdot 14

14

divides by

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 7

= 2^{2} \cdot 7

= 2^{2} \cdot 7

Apply radical rule:

= 7 2^{2}

Apply radical rule:

2^{2} = 2

= 27

u_{1, 2} = \frac{- ( - 2 ) \pm 2 7}{2 ( - 1 )}

Separate the solutions

u_{1} = \frac{- ( - 2 ) + 2 7}{2 ( - 1 )}, u_{2} = \frac{- ( - 2 ) - 2 7}{2 ( - 1 )}

u = \frac{- ( - 2 ) + 2 7}{2 ( - 1 )} : - 1 - 7

\frac{- ( - 2 ) + 2 7}{2 ( - 1 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{2 + 2 7}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 + 2 7}{- 2}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{2 + 2 7}{2}

Cancel

\frac{2 + 2 7}{2} : 1 + 7

\frac{2 + 2 7}{2}

Factor

2 + 27 : 2 (1 + 7)

2 + 27

Rewrite as

= 2 \cdot 1 + 27

Factor out common term

2

= 2 (1 + 7)

= \frac{2 ( 1 + 7 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 1 + 7

= - (1 + 7)

Distribute parentheses

= - (1) - (7)

Apply minus-plus rules

+ (- a) = - a

= - 1 - 7

u = \frac{- ( - 2 ) - 2 7}{2 ( - 1 )} : 7 - 1

\frac{- ( - 2 ) - 2 7}{2 ( - 1 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{2 - 2 7}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2 - 2 7}{- 2}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

2 - 27 = - (27 - 2)

= \frac{2 7 - 2}{2}

Factor

27 - 2 : 2 (7 - 1)

27 - 2

Rewrite as

= 27 - 2 \cdot 1

Factor out common term

2

= 2 (7 - 1)

= \frac{2 ( 7 - 1 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 7 - 1

The solutions to the quadratic equation are:

u = - 1 - 7, u = 7 - 1

Substitute back

u = sec (y)

sec (y) = - 1 - 7, sec (y) = 7 - 1

sec (y) = - 1 - 7, sec (y) = 7 - 1

sec (y) = - 1 - 7 : y = arcsec (- 1 - 7) + 2 πn, y = - arcsec (- 1 - 7) + 2 πn

sec (y) = - 1 - 7

Apply trig inverse properties

sec (y) = - 1 - 7

General solutions for

sec (y) = - 1 - 7

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

y = arcsec (- 1 - 7) + 2 πn, y = - arcsec (- 1 - 7) + 2 πn

y = arcsec (- 1 - 7) + 2 πn, y = - arcsec (- 1 - 7) + 2 πn

sec (y) = 7 - 1 : y = arcsec (7 - 1) + 2 πn, y = 2 π - arcsec (7 - 1) + 2 πn

sec (y) = 7 - 1

Apply trig inverse properties

sec (y) = 7 - 1

General solutions for

sec (y) = 7 - 1

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

y = arcsec (7 - 1) + 2 πn, y = 2 π - arcsec (7 - 1) + 2 πn

y = arcsec (7 - 1) + 2 πn, y = 2 π - arcsec (7 - 1) + 2 πn

Combine all the solutions

y = arcsec (- 1 - 7) + 2 πn, y = - arcsec (- 1 - 7) + 2 πn, y = arcsec (7 - 1) + 2 πn, y = 2 π - arcsec (7 - 1) + 2 πn

Show solutions in decimal form

y = 1.84864 \dots + 2 πn, y = - 1.84864 \dots + 2 πn, y = 0.91772 \dots + 2 πn, y = 2 π - 0.91772 \dots + 2 πn

Graph

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