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3sec^2(x)+4cos^2(x)=7

Solution

3 sec^{2} (x) + 4 cos^{2} (x) = 7

Solution

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = 2 πn, x = π + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

3 sec^{2} (x) + 4 cos^{2} (x) = 7

Subtract

7

from both sides

3 sec^{2} (x) + 4 cos^{2} (x) - 7 = 0

Rewrite using trig identities

- 7 + 3 sec^{2} (x) + 4 cos^{2} (x)

Use the basic trigonometric identity:

cos (x) = \frac{1}{sec ( x )}

= - 7 + 3 sec^{2} (x) + 4 (\frac{1}{sec ( x )})^{2}

4 (\frac{1}{sec ( x )})^{2} = \frac{4}{sec ^{2} ( x )}

4 (\frac{1}{sec ( x )})^{2}

(\frac{1}{sec ( x )})^{2} = \frac{1}{sec ^{2} ( x )}

(\frac{1}{sec ( x )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{sec ^{2} ( x )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{sec ^{2} ( x )}

= 4 \cdot \frac{1}{sec ^{2} ( x )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 4}{sec ^{2} ( x )}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{sec ^{2} ( x )}

= - 7 + 3 sec^{2} (x) + \frac{4}{sec ^{2} ( x )}

- 7 + \frac{4}{sec ^{2} ( x )} + 3 sec^{2} (x) = 0

Solve by substitution

- 7 + \frac{4}{sec ^{2} ( x )} + 3 sec^{2} (x) = 0

Let:

sec (x) = u

- 7 + \frac{4}{u ^{2}} + 3 u^{2} = 0

- 7 + \frac{4}{u ^{2}} + 3 u^{2} = 0 : u = \frac{2 3}{3}, u = - \frac{2 3}{3}, u = 1, u = - 1

- 7 + \frac{4}{u ^{2}} + 3 u^{2} = 0

Multiply both sides by

u^{2}

- 7 + \frac{4}{u ^{2}} + 3 u^{2} = 0

Multiply both sides by

u^{2}

- 7 u^{2} + \frac{4}{u ^{2}} u^{2} + 3 u^{2} u^{2} = 0 \cdot u^{2}

Simplify

- 7 u^{2} + \frac{4}{u ^{2}} u^{2} + 3 u^{2} u^{2} = 0 \cdot u^{2}

Simplify

\frac{4}{u ^{2}} u^{2} : 4

\frac{4}{u ^{2}} u^{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{4 u ^{2}}{u ^{2}}

Cancel the common factor:

u^{2}

= 4

Simplify

3 u^{2} u^{2} : 3 u^{4}

3 u^{2} u^{2}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

u^{2} u^{2} = u^{2 + 2}

= 3 u^{2 + 2}

Add the numbers:

2 + 2 = 4

= 3 u^{4}

Simplify

0 \cdot u^{2} : 0

0 \cdot u^{2}

Apply rule

0 \cdot a = 0

= 0

- 7 u^{2} + 4 + 3 u^{4} = 0

- 7 u^{2} + 4 + 3 u^{4} = 0

- 7 u^{2} + 4 + 3 u^{4} = 0

Solve

- 7 u^{2} + 4 + 3 u^{4} = 0 : u = \frac{2 3}{3}, u = - \frac{2 3}{3}, u = 1, u = - 1

- 7 u^{2} + 4 + 3 u^{4} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

3 u^{4} - 7 u^{2} + 4 = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

3 v^{2} - 7 v + 4 = 0

Solve

3 v^{2} - 7 v + 4 = 0 : v = \frac{4}{3}, v = 1

3 v^{2} - 7 v + 4 = 0

Solve with the quadratic formula

3 v^{2} - 7 v + 4 = 0

Quadratic Equation Formula:

For

a = 3, b = - 7, c = 4

v_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 \cdot 4}{2 \cdot 3}

v_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 \cdot 4}{2 \cdot 3}

(- 7)^{2} - 4 \cdot 3 \cdot 4 = 1

(- 7)^{2} - 4 \cdot 3 \cdot 4

Apply exponent rule:

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

n

is even

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 3 \cdot 4

Multiply the numbers:

4 \cdot 3 \cdot 4 = 48

= 7^{2} - 48

7^{2} = 49

= 49 - 48

Subtract the numbers:

49 - 48 = 1

= 1

Apply rule

1 = 1

= 1

v_{1, 2} = \frac{- ( - 7 ) \pm 1}{2 \cdot 3}

Separate the solutions

v_{1} = \frac{- ( - 7 ) + 1}{2 \cdot 3}, v_{2} = \frac{- ( - 7 ) - 1}{2 \cdot 3}

v = \frac{- ( - 7 ) + 1}{2 \cdot 3} : \frac{4}{3}

\frac{- ( - 7 ) + 1}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{7 + 1}{2 \cdot 3}

Add the numbers:

7 + 1 = 8

= \frac{8}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{8}{6}

Cancel the common factor:

2

= \frac{4}{3}

v = \frac{- ( - 7 ) - 1}{2 \cdot 3} : 1

\frac{- ( - 7 ) - 1}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{7 - 1}{2 \cdot 3}

Subtract the numbers:

7 - 1 = 6

= \frac{6}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6}{6}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

v = \frac{4}{3}, v = 1

v = \frac{4}{3}, v = 1

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = \frac{4}{3} : u = \frac{2 3}{3}, u = - \frac{2 3}{3}

u^{2} = \frac{4}{3}

For

x^{2} = f (a)

the solutions are

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

u = \frac{4}{3}, u = - \frac{4}{3}

\frac{4}{3} = \frac{2 3}{3}

\frac{4}{3}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{4}{3}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2}{3}

Rationalize

\frac{2}{3} : \frac{2 3}{3}

\frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{2 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{2 3}{3}

= \frac{2 3}{3}

- \frac{4}{3} = - \frac{2 3}{3}

- \frac{4}{3}

Simplify

\frac{4}{3} : \frac{2}{3}

\frac{4}{3}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{4}{3}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2}{3}

= - \frac{2}{3}

Rationalize

- \frac{2}{3} : - \frac{2 3}{3}

- \frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{2 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{2 3}{3}

= - \frac{2 3}{3}

u = \frac{2 3}{3}, u = - \frac{2 3}{3}

Solve

u^{2} = 1 : u = 1, u = - 1

u^{2} = 1

For

x^{2} = f (a)

the solutions are

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

u = 1, u = - 1

1 = 1

1

Apply rule

1 = 1

= 1

- 1 = - 1

- 1

Apply rule

1 = 1

= - 1

u = 1, u = - 1

The solutions are

u = \frac{2 3}{3}, u = - \frac{2 3}{3}, u = 1, u = - 1

u = \frac{2 3}{3}, u = - \frac{2 3}{3}, u = 1, u = - 1

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

- 7 + \frac{4}{u ^{2}} + 3 u^{2}

and compare to zero

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{2 3}{3}, u = - \frac{2 3}{3}, u = 1, u = - 1

Substitute back

u = sec (x)

sec (x) = \frac{2 3}{3}, sec (x) = - \frac{2 3}{3}, sec (x) = 1, sec (x) = - 1

sec (x) = \frac{2 3}{3}, sec (x) = - \frac{2 3}{3}, sec (x) = 1, sec (x) = - 1

sec (x) = \frac{2 3}{3} : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sec (x) = \frac{2 3}{3}

General solutions for

sec (x) = \frac{2 3}{3}

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}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sec (x) = - \frac{2 3}{3} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

sec (x) = - \frac{2 3}{3}

General solutions for

sec (x) = - \frac{2 3}{3}

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{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

sec (x) = 1 : x = 2 πn

sec (x) = 1

General solutions for

sec (x) = 1

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 = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

sec (x) = - 1 : x = π + 2 πn

sec (x) = - 1

General solutions for

sec (x) = - 1

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 = π + 2 πn

x = π + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = 2 πn, x = π + 2 πn

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