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Popular >

solvefor x,cos^2(x)+cos^2(y)cos^2(z)=1

Solution

solve for

x, cos^{2} (x) + cos^{2} (y) cos^{2} (z) = 1

Solution

x = arcsin (- cos (y) cos (z)) + 2 πn, x = π + arcsin (cos (y) cos (z)) + 2 πn, x = arcsin (cos (y) cos (z)) + 2 πn, x = π + arcsin (- cos (y) cos (z)) + 2 πn

Solution steps

cos^{2} (x) + cos^{2} (y) cos^{2} (z) = 1

Subtract

1

from both sides

cos^{2} (x) + cos^{2} (y) cos^{2} (z) - 1 = 0

Rewrite using trig identities

- 1 + cos^{2} (x) + cos^{2} (y) cos^{2} (z)

Use the Pythagorean identity:

1 = cos^{2} (x) + sin^{2} (x)

1 - cos^{2} (x) = sin^{2} (x)

= cos^{2} (y) cos^{2} (z) - sin^{2} (x)

- sin^{2} (x) + cos^{2} (y) cos^{2} (z) = 0

Factor

- sin^{2} (x) + cos^{2} (y) cos^{2} (z) : (cos (y) cos (z) + sin (x)) (cos (y) cos (z) - sin (x))

- sin^{2} (x) + cos^{2} (y) cos^{2} (z)

Rewrite

cos^{2} (y) cos^{2} (z)

(cos (y) cos (z))^{2}

cos^{2} (y) cos^{2} (z)

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

cos^{2} (y) cos^{2} (z) = (cos (y) cos (z))^{2}

= (cos (y) cos (z))^{2}

= - sin^{2} (x) + (cos (y) cos (z))^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(cos (y) cos (z))^{2} - sin^{2} (x) = (cos (y) cos (z) + sin (x)) (cos (y) cos (z) - sin (x))

= (cos (y) cos (z) + sin (x)) (cos (y) cos (z) - sin (x))

(cos (y) cos (z) + sin (x)) (cos (y) cos (z) - sin (x)) = 0

Solving each part separately

cos (y) cos (z) + sin (x) = 0 or cos (y) cos (z) - sin (x) = 0

cos (y) cos (z) + sin (x) = 0 : x = arcsin (- cos (y) cos (z)) + 2 πn, x = π + arcsin (cos (y) cos (z)) + 2 πn

cos (y) cos (z) + sin (x) = 0

Move

cos (y) cos (z)

to the right side

cos (y) cos (z) + sin (x) = 0

Subtract

cos (y) cos (z)

from both sides

cos (y) cos (z) + sin (x) - cos (y) cos (z) = 0 - cos (y) cos (z)

Simplify

sin (x) = - cos (y) cos (z)

sin (x) = - cos (y) cos (z)

Apply trig inverse properties

sin (x) = - cos (y) cos (z)

General solutions for

sin (x) = - cos (y) cos (z)

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- cos (y) cos (z)) + 2 πn, x = π + arcsin (cos (y) cos (z)) + 2 πn

x = arcsin (- cos (y) cos (z)) + 2 πn, x = π + arcsin (cos (y) cos (z)) + 2 πn

cos (y) cos (z) - sin (x) = 0 : x = arcsin (cos (y) cos (z)) + 2 πn, x = π + arcsin (- cos (y) cos (z)) + 2 πn

cos (y) cos (z) - sin (x) = 0

Move

cos (y) cos (z)

to the right side

cos (y) cos (z) - sin (x) = 0

Subtract

cos (y) cos (z)

from both sides

cos (y) cos (z) - sin (x) - cos (y) cos (z) = 0 - cos (y) cos (z)

Simplify

- sin (x) = - cos (y) cos (z)

- sin (x) = - cos (y) cos (z)

Divide both sides by

- 1

- sin (x) = - cos (y) cos (z)

Divide both sides by

- 1

\frac{- sin ( x )}{- 1} = \frac{- cos ( y ) cos ( z )}{- 1}

Simplify

sin (x) = cos (y) cos (z)

sin (x) = cos (y) cos (z)

Apply trig inverse properties

sin (x) = cos (y) cos (z)

General solutions for

sin (x) = cos (y) cos (z)

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (cos (y) cos (z)) + 2 πn, x = π + arcsin (- cos (y) cos (z)) + 2 πn

x = arcsin (cos (y) cos (z)) + 2 πn, x = π + arcsin (- cos (y) cos (z)) + 2 πn

Combine all the solutions

x = arcsin (- cos (y) cos (z)) + 2 πn, x = π + arcsin (cos (y) cos (z)) + 2 πn, x = arcsin (cos (y) cos (z)) + 2 πn, x = π + arcsin (- cos (y) cos (z)) + 2 πn

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