What is the general solution for sqrt(1-cos^2(x))=0.75sqrt(1-cos^2(y)) ?

The general solution for sqrt(1-cos^2(x))=0.75sqrt(1-cos^2(y)) is

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sqrt(1-cos^2(x))=0.75sqrt(1-cos^2(y))

Solution

1 - cos^{2} (x) = 0.75 1 - cos^{2} (y)

Solution

x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

Solution steps

1 - cos^{2} (x) = 0.75 1 - cos^{2} (y)

Solve by substitution

1 - cos^{2} (x) = 0.75 1 - cos^{2} (y)

Let:

cos (x) = u

1 - u^{2} = 0.75 1 - cos^{2} (y)

1 - u^{2} = 0.75 1 - cos^{2} (y) : u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

1 - u^{2} = 0.75 1 - cos^{2} (y)

Square both sides:

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

1 - u^{2} = 0.75 1 - cos^{2} (y)

(1 - u^{2})^{2} = (0.75 1 - cos^{2} (y))^{2}

Expand

(1 - u^{2})^{2} : 1 - u^{2}

(1 - u^{2})^{2}

Apply radical rule:

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

= ((1 - u^{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

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

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 1 - u^{2}

Expand

(0.75 1 - cos^{2} (y))^{2} : 0.5625 - 0.5625 cos^{2} (y)

(0.75 1 - cos^{2} (y))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 0.7 5^{2} (- cos^{2} (y) + 1)^{2}

(1 - cos^{2} (y))^{2} : 1 - cos^{2} (y)

Apply radical rule:

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

= ((1 - cos^{2} (y))^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (1 - cos^{2} (y))^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 1 - cos^{2} (y)

= 0.7 5^{2} (1 - cos^{2} (y))

0.7 5^{2} = 0.5625

= 0.5625 (- cos^{2} (y) + 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 0.5625, b = 1, c = cos^{2} (y)

= 0.5625 \cdot 1 - 0.5625 cos^{2} (y)

= 1 \cdot 0.5625 - 0.5625 cos^{2} (y)

Multiply the numbers:

1 \cdot 0.5625 = 0.5625

= 0.5625 - 0.5625 cos^{2} (y)

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

Solve

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y) : u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

Move

1

to the right side

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

Subtract

1

from both sides

1 - u^{2} - 1 = 0.5625 - 0.5625 cos^{2} (y) - 1

Simplify

- u^{2} = - 0.5625 cos^{2} (y) - 0.4375

- u^{2} = - 0.5625 cos^{2} (y) - 0.4375

Divide both sides by

- 1

- u^{2} = - 0.5625 cos^{2} (y) - 0.4375

Divide both sides by

- 1

\frac{- u ^{2}}{- 1} = - \frac{0.5625 cos ^{2} ( y )}{- 1} - \frac{0.4375}{- 1}

Simplify

\frac{- u ^{2}}{- 1} = - \frac{0.5625 cos ^{2} ( y )}{- 1} - \frac{0.4375}{- 1}

Simplify

\frac{- u ^{2}}{- 1} : u^{2}

\frac{- u ^{2}}{- 1}

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= u^{2}

Simplify

- \frac{0.5625 cos ^{2} ( y )}{- 1} - \frac{0.4375}{- 1} : 0.5625 cos^{2} (y) + 0.4375

- \frac{0.5625 cos ^{2} ( y )}{- 1} - \frac{0.4375}{- 1}

Apply rule

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

= \frac{- 0.5625 cos ^{2} ( y ) - 0.4375}{- 1}

Apply the fraction rule:

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

= - \frac{- 0.5625 cos ^{2} ( y ) - 0.4375}{1}

Apply rule

\frac{a}{1} = a

= - (- 0.5625 cos^{2} (y) - 0.4375)

Distribute parentheses

= - (- 0.5625 cos^{2} (y)) - (- 0.4375)

Apply minus-plus rules

- (- a) = a

= 0.5625 cos^{2} (y) + 0.4375

u^{2} = 0.5625 cos^{2} (y) + 0.4375

u^{2} = 0.5625 cos^{2} (y) + 0.4375

u^{2} = 0.5625 cos^{2} (y) + 0.4375

For

x^{2} = f (a)

the solutions are

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

u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

Verify Solutions:

u = 0.5625 cos^{2} (y) + 0.4375

True

, u = - 0.5625 cos^{2} (y) + 0.4375

True

Check the solutions by plugging them into

1 - u^{2} = 0.75 1 - cos^{2} (y)

Remove the ones that don't agree with the equation.

Plug in

u = 0.5625 cos^{2} (y) + 0.4375 :

True

1 - (0.5625 cos^{2} (y) + 0.4375)^{2} = 0.75 1 - cos^{2} (y)

Simplify

1 - (0.5625 cos^{2} (y) + 0.4375)^{2} : - 0.5625 cos^{2} (y) + 0.5625

1 - (0.5625 cos^{2} (y) + 0.4375)^{2}

(0.5625 cos^{2} (y) + 0.4375)^{2} = 0.5625 cos^{2} (y) + 0.4375

(0.5625 cos^{2} (y) + 0.4375)^{2}

Apply radical rule:

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

= ((0.5625 cos^{2} (y) + 0.4375)^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (0.5625 cos^{2} (y) + 0.4375)^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 0.5625 cos^{2} (y) + 0.4375

= 1 - (0.5625 cos^{2} (y) + 0.4375)

Expand

1 - (0.5625 cos^{2} (y) + 0.4375) : - 0.5625 cos^{2} (y) + 0.5625

1 - (0.5625 cos^{2} (y) + 0.4375)

- (0.5625 cos^{2} (y) + 0.4375) : - 0.5625 cos^{2} (y) - 0.4375

- (0.5625 cos^{2} (y) + 0.4375)

Distribute parentheses

= - (0.5625 cos^{2} (y)) - (0.4375)

Apply minus-plus rules

+ (- a) = - a

= - 0.5625 cos^{2} (y) - 0.4375

= 1 - 0.5625 cos^{2} (y) - 0.4375

Subtract the numbers:

1 - 0.4375 = 0.5625

= - 0.5625 cos^{2} (y) + 0.5625

= - 0.5625 cos^{2} (y) + 0.5625

- 0.5625 cos^{2} (y) + 0.5625 = 0.75 1 - cos^{2} (y)

True

Plug in

u = - 0.5625 cos^{2} (y) + 0.4375 :

True

1 - (- 0.5625 cos^{2} (y) + 0.4375)^{2} = 0.75 1 - cos^{2} (y)

Simplify

1 - (- 0.5625 cos^{2} (y) + 0.4375)^{2} : - 0.5625 cos^{2} (y) + 0.5625

1 - (- 0.5625 cos^{2} (y) + 0.4375)^{2}

(- 0.5625 cos^{2} (y) + 0.4375)^{2} = 0.5625 cos^{2} (y) + 0.4375

(- 0.5625 cos^{2} (y) + 0.4375)^{2}

Apply exponent rule:

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

n

is even

(- 0.5625 cos^{2} (y) + 0.4375)^{2} = (0.5625 cos^{2} (y) + 0.4375)^{2}

= (0.5625 cos^{2} (y) + 0.4375)^{2}

Apply radical rule:

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

= ((0.5625 cos^{2} (y) + 0.4375)^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= (0.5625 cos^{2} (y) + 0.4375)^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 0.5625 cos^{2} (y) + 0.4375

= 1 - (0.5625 cos^{2} (y) + 0.4375)

Expand

1 - (0.5625 cos^{2} (y) + 0.4375) : - 0.5625 cos^{2} (y) + 0.5625

1 - (0.5625 cos^{2} (y) + 0.4375)

- (0.5625 cos^{2} (y) + 0.4375) : - 0.5625 cos^{2} (y) - 0.4375

- (0.5625 cos^{2} (y) + 0.4375)

Distribute parentheses

= - (0.5625 cos^{2} (y)) - (0.4375)

Apply minus-plus rules

+ (- a) = - a

= - 0.5625 cos^{2} (y) - 0.4375

= 1 - 0.5625 cos^{2} (y) - 0.4375

Subtract the numbers:

1 - 0.4375 = 0.5625

= - 0.5625 cos^{2} (y) + 0.5625

= - 0.5625 cos^{2} (y) + 0.5625

- 0.5625 cos^{2} (y) + 0.5625 = 0.75 1 - cos^{2} (y)

True

The solutions are

u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

Substitute back

u = cos (x)

cos (x) = 0.5625 cos^{2} (y) + 0.4375, cos (x) = - 0.5625 cos^{2} (y) + 0.4375

cos (x) = 0.5625 cos^{2} (y) + 0.4375, cos (x) = - 0.5625 cos^{2} (y) + 0.4375

cos (x) = 0.5625 cos^{2} (y) + 0.4375 : x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn

cos (x) = 0.5625 cos^{2} (y) + 0.4375

Apply trig inverse properties

cos (x) = 0.5625 cos^{2} (y) + 0.4375

General solutions for

cos (x) = 0.5625 cos^{2} (y) + 0.4375

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

x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn

x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn

cos (x) = - 0.5625 cos^{2} (y) + 0.4375 : x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

cos (x) = - 0.5625 cos^{2} (y) + 0.4375

Apply trig inverse properties

cos (x) = - 0.5625 cos^{2} (y) + 0.4375

General solutions for

cos (x) = - 0.5625 cos^{2} (y) + 0.4375

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

x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

Combine all the solutions

x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(1-cos^2(x))=0.75sqrt(1-cos^2(y)) ?
The general solution for sqrt(1-cos^2(x))=0.75sqrt(1-cos^2(y)) is