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Popular >

4cos(θ)-3sec(θ)=0

Solution

4 cos (θ) - 3 sec (θ) = 0

Solution

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

Degrees

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

Solution steps

4 cos (θ) - 3 sec (θ) = 0

Rewrite using trig identities

- 3 sec (θ) + 4 cos (θ)

Use the basic trigonometric identity:

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

= - 3 sec (θ) + 4 \cdot \frac{1}{sec ( θ )}

4 \cdot \frac{1}{sec ( θ )} = \frac{4}{sec ( θ )}

4 \cdot \frac{1}{sec ( θ )}

Multiply fractions:

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

= \frac{1 \cdot 4}{sec ( θ )}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{sec ( θ )}

= - 3 sec (θ) + \frac{4}{sec ( θ )}

\frac{4}{sec ( θ )} - 3 sec (θ) = 0

Solve by substitution

\frac{4}{sec ( θ )} - 3 sec (θ) = 0

Let:

sec (θ) = u

\frac{4}{u} - 3 u = 0

\frac{4}{u} - 3 u = 0 : u = \frac{2 3}{3}, u = - \frac{2 3}{3}

\frac{4}{u} - 3 u = 0

Multiply both sides by

u

\frac{4}{u} - 3 u = 0

Multiply both sides by

u

\frac{4}{u} u - 3 uu = 0 \cdot u

Simplify

\frac{4}{u} u - 3 uu = 0 \cdot u

Simplify

\frac{4}{u} u : 4

\frac{4}{u} u

Multiply fractions:

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

= \frac{4 u}{u}

Cancel the common factor:

u

= 4

Simplify

- 3 uu : - 3 u^{2}

- 3 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= - 3 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= - 3 u^{2}

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

4 - 3 u^{2} = 0

4 - 3 u^{2} = 0

4 - 3 u^{2} = 0

Solve

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

4 - 3 u^{2} = 0

Move

4

to the right side

4 - 3 u^{2} = 0

Subtract

4

from both sides

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

Simplify

- 3 u^{2} = - 4

- 3 u^{2} = - 4

Divide both sides by

- 3

- 3 u^{2} = - 4

Divide both sides by

- 3

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

Simplify

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

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

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{4}{u} - 3 u

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

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

Substitute back

u = sec (θ)

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

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

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

sec (θ) = \frac{2 3}{3}

General solutions for

sec (θ) = \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}

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

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

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

sec (θ) = - \frac{2 3}{3}

General solutions for

sec (θ) = - \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}

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

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

Combine all the solutions

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

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