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Popular >

cos^4(x)=0.37

Solution

cos^{4} (x) = 0.37

Solution

x = 0.67625 \dots + 2 πn, x = 2 π - 0.67625 \dots + 2 πn, x = 2.46533 \dots + 2 πn, x = - 2.46533 \dots + 2 πn

Degrees

x = 38.74668 \dots^{\circ} + 36 0^{\circ} n, x = 321.25331 \dots^{\circ} + 36 0^{\circ} n, x = 141.25331 \dots^{\circ} + 36 0^{\circ} n, x = - 141.25331 \dots^{\circ} + 36 0^{\circ} n

Solution steps

cos^{4} (x) = 0.37

Solve by substitution

cos^{4} (x) = 0.37

Let:

cos (x) = u

u^{4} = 0.37

u^{4} = 0.37

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

v^{2} = 0.37

Solve

v^{2} = 0.37 : v = 0.37, v = - 0.37

v^{2} = 0.37

For

(g (x))^{2} = f (a)

the solutions are

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

v = 0.37, v = - 0.37

v = 0.37, v = - 0.37

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = 0.37

For

x^{2} = f (a)

the solutions are

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

u = 0.37, u = - 0.37

0.37

Apply radical rule:

a = a^{\frac{1}{2}}

= (0.3 7^{\frac{1}{2}})^{\frac{1}{2}}

Apply exponent rule:

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

= 0.3 7^{\frac{1}{2} \cdot \frac{1}{2}}

\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

\frac{1}{2} \cdot \frac{1}{2}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

= 0.3 7^{\frac{1}{4}}

- 0.37

Simplify

0.37

Apply radical rule:

a = a^{\frac{1}{2}}

= (0.3 7^{\frac{1}{2}})^{\frac{1}{2}}

Apply exponent rule:

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

= 0.3 7^{\frac{1}{2} \cdot \frac{1}{2}}

\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

\frac{1}{2} \cdot \frac{1}{2}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

= 0.3 7^{\frac{1}{4}}

Solve

u^{2} = - 0.37

For

x^{2} = f (a)

the solutions are

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

u = - 0.37, u = - - 0.37

Simplify

- 0.37

Apply radical rule:

- a = - 1 a

- 0.37 = - 1 0.37

= - 1 0.37

Apply imaginary number rule:

- 1 = i

= i 0.37

Apply radical rule:

a = a^{\frac{1}{2}}

= (0.3 7^{\frac{1}{2}})^{\frac{1}{2}}

Apply exponent rule:

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

= 0.3 7^{\frac{1}{2} \cdot \frac{1}{2}}

\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

\frac{1}{2} \cdot \frac{1}{2}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

= 0.3 7^{\frac{1}{4}}

Simplify

- - 0.37

Simplify

- 0.37

Apply radical rule:

- a = - 1 a

- 0.37 = - 1 0.37

= - 1 0.37

Apply imaginary number rule:

- 1 = i

= i 0.37

Apply radical rule:

a = a^{\frac{1}{2}}

= (0.3 7^{\frac{1}{2}})^{\frac{1}{2}}

Apply exponent rule:

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

= 0.3 7^{\frac{1}{2} \cdot \frac{1}{2}}

\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

\frac{1}{2} \cdot \frac{1}{2}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

= 0.3 7^{\frac{1}{4}}

The solutions are

Substitute back

u = cos (x)

Apply trig inverse properties

General solutions for

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

Apply trig inverse properties

General solutions for

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

No Solution

No Solution

No Solution

No Solution

Combine all the solutions

Show solutions in decimal form

x = 0.67625 \dots + 2 πn, x = 2 π - 0.67625 \dots + 2 πn, x = 2.46533 \dots + 2 πn, x = - 2.46533 \dots + 2 πn

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