Question

2. Find the general relation of the equation cos3A+cos5A=0

Answer 1

Answer:

$A=\frac{\pi}{8}+\frac{n\pi}{4}or\ A=\frac{\pi}{2}+n\pi$

Step-by-step explanation:

Find angles

$cos3A+cos5A=0$

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Transform the expression using the sum-to-product formula

$2cos(\frac{3A+5A}{2})cos(\frac{3A-5A}{2})=0$

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Combine like terms

$2cos(\frac{8A}{2})cos(\frac{3A-5A}{2})=0\\\\ 2cos(\frac{8A}{2})cos(\frac{-2A}{2})=0$

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Divide both sides of the equation by the coefficient of variable

$cos(\frac{8A}{2})cos(\frac{-2A}{2})=0$

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Apply zero product property that at least one factor is zero

$cos(\frac{8A}{2})=0\ or\ cos(\frac{-2A}{2})=0$

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Cos (8A/2) = 0:

Cross out the common factor

$cos\ 4A=0$

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Solve the trigonometric equation to find a particular solution

$4A=\frac{\pi}{2}or\ 4A=\frac{3\pi}{2}$

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Solve the trigonometric equation to find a general solution

$4A=\frac{\pi}{2}+2n\pi \ or\\ \\ 4A=\frac{3 \pi}{2}+2n \pi\\ \\A=\frac{\pi}{8}+\frac{n \pi}{4\\}$

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cos(-2A/2) = 0

Reduce the fraction

$cos(-A)=0$

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Simplify the expression using the symmetry of trigonometric function

$cosA=0$

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Solve the trigonometric equation to find a particular solution

$A=\frac{\pi }{2}\ or\ A=\frac{3 \pi}{2}$

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Solve the trigonometric equation to find a general solution

$A=\frac{\pi}{2}+2n\pi\ or\ A=\frac{3\pi}{2}+2n\pi,n\in\ Z$

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Find the union of solution sets

$A=\frac{\pi}{2}+n\pi$

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A = π/8 + nπ/4 or A = π/2 + nπ, n ∈ Z

Find the union of solution sets

$A=\frac{\pi}{8}+\frac{n\pi}{4}\ or\ A=\frac{\pi}{2}+n\pi ,n\in Z$

I hope this helps you

:)