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Popular >

16^{sin(x)}=8^{csc(x)}

Solution

1 6^{s i n (x)} = 8^{c s c (x)}

Solution

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

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

1 6^{s i n (x)} = 8^{c s c (x)}

Subtract

8^{c s c (x)}

from both sides

1 6^{s i n (x)} - 8^{c s c (x)} = 0

Rewrite using trig identities

1 6^{s i n (x)} - 8^{c s c (x)}

Use the basic trigonometric identity:

sin (x) = \frac{1}{csc ( x )}

= 1 6^{\frac{1}{c s c ( x )}} - 8^{c s c (x)}

1 6^{\frac{1}{c s c ( x )}} - 8^{c s c (x)} = 0

Solve by substitution

1 6^{\frac{1}{c s c ( x )}} - 8^{c s c (x)} = 0

Let:

csc (x) = u

1 6^{\frac{1}{u}} - 8^{u} = 0

1 6^{\frac{1}{u}} - 8^{u} = 0 : u = \frac{2 3}{3}, u = - \frac{2 3}{3}

1 6^{\frac{1}{u}} - 8^{u} = 0

Add

8^{u}

to both sides

1 6^{\frac{1}{u}} - 8^{u} + 8^{u} = 0 + 8^{u}

Simplify

1 6^{\frac{1}{u}} = 8^{u}

Apply exponent rules

1 6^{\frac{1}{u}} = 8^{u}

Convert to base

2 : 2^{4 \cdot \frac{1}{u}} = 2^{3 u}

Convert

8

to base

2

8 = 2^{3}

1 6^{\frac{1}{u}} = (2^{3})^{u}

Convert

16

to base

2

16 = 2^{4}

(2^{4})^{\frac{1}{u}} = (2^{3})^{u}

Apply exponent rule:

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

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

2^{4 \cdot \frac{1}{u}} = (2^{3})^{u}

Apply exponent rule:

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

(2^{3})^{u} = 2^{3 u}

2^{4 \cdot \frac{1}{u}} = 2^{3 u}

2^{4 \cdot \frac{1}{u}} = 2^{3 u}

a^{f (x)} = a^{g (x)}

, then

f (x) = g (x)

4 \cdot \frac{1}{u} = 3 u

Simplify

\frac{4}{u} = 3 u

\frac{4}{u} = 3 u

Solve

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

\frac{4}{u} = 3 u

Multiply both sides by

u

\frac{4}{u} = 3 u

Multiply both sides by

u

\frac{4}{u} u = 3 uu

Simplify

3 uu : 3 u^{2}

\frac{4}{u} u = 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}

4 = 3 u^{2}

4 = 3 u^{2}

Solve

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

4 = 3 u^{2}

Switch sides

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}

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}

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

Substitute back

u = csc (x)

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

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

csc (x) = \frac{2 3}{3} : x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

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

General solutions for

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

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

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

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

General solutions for

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

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

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

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

Combine all the solutions

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

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