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Popular >

sin^4(x/3)+cos^4(x/3)= 5/8

Solution

sin^{4} (\frac{x}{3}) + cos^{4} (\frac{x}{3}) = \frac{5}{8}

Solution

x = \frac{π}{2} + 6 πn, x = \frac{11 π}{2} + 6 πn, x = \frac{5 π}{2} + 6 πn, x = \frac{7 π}{2} + 6 πn, x = π + 6 πn, x = 5 π + 6 πn, x = 2 π + 6 πn, x = 4 π + 6 πn

+1

Solution steps

sin^{4} (\frac{x}{3}) + cos^{4} (\frac{x}{3}) = \frac{5}{8}

Subtract

\frac{5}{8}

from both sides

sin^{4} (\frac{x}{3}) + cos^{4} (\frac{x}{3}) - \frac{5}{8} = 0

Simplify

sin^{4} (\frac{x}{3}) + cos^{4} (\frac{x}{3}) - \frac{5}{8} : \frac{8 sin ^{4} ( \frac{x}{3} ) + 8 cos ^{4} ( \frac{x}{3} ) - 5}{8}

\frac{8 sin ^{4} ( \frac{x}{3} ) + 8 cos ^{4} ( \frac{x}{3} ) - 5}{8} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

8 sin^{4} (\frac{x}{3}) + 8 cos^{4} (\frac{x}{3}) - 5 = 0

Let:

u = \frac{x}{3}

8 sin^{4} (u) + 8 cos^{4} (u) - 5 = 0

Apply exponent rule:

a^{b} = a^{2} a^{b - 2}

- 5 + 8 cos^{4} (u) + 8 sin^{2} (u) sin^{2} (u) = 0

Rewrite using trig identities

3 - 16 cos^{2} (u) + 16 cos^{4} (u) = 0

Solve by substitution

cos (u) = \frac{3}{2}, cos (u) = - \frac{3}{2}, cos (u) = \frac{1}{2}, cos (u) = - \frac{1}{2}

cos (u) = \frac{3}{2} : u = \frac{π}{6} + 2 πn, u = \frac{11 π}{6} + 2 πn

cos (u) = - \frac{3}{2} : u = \frac{5 π}{6} + 2 πn, u = \frac{7 π}{6} + 2 πn

cos (u) = \frac{1}{2} : u = \frac{π}{3} + 2 πn, u = \frac{5 π}{3} + 2 πn

cos (u) = - \frac{1}{2} : u = \frac{2 π}{3} + 2 πn, u = \frac{4 π}{3} + 2 πn

Combine all the solutions

u = \frac{π}{6} + 2 πn, u = \frac{11 π}{6} + 2 πn, u = \frac{5 π}{6} + 2 πn, u = \frac{7 π}{6} + 2 πn, u = \frac{π}{3} + 2 πn, u = \frac{5 π}{3} + 2 πn, u = \frac{2 π}{3} + 2 πn, u = \frac{4 π}{3} + 2 πn

Substitute back

u = \frac{x}{3}

\frac{x}{3} = \frac{π}{6} + 2 πn : x = \frac{π}{2} + 6 πn

\frac{x}{3} = \frac{11 π}{6} + 2 πn : x = \frac{11 π}{2} + 6 πn

\frac{x}{3} = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{2} + 6 πn

\frac{x}{3} = \frac{7 π}{6} + 2 πn : x = \frac{7 π}{2} + 6 πn

\frac{x}{3} = \frac{π}{3} + 2 πn : x = π + 6 πn

\frac{x}{3} = \frac{5 π}{3} + 2 πn : x = 5 π + 6 πn

\frac{x}{3} = \frac{2 π}{3} + 2 πn : x = 2 π + 6 πn

\frac{x}{3} = \frac{4 π}{3} + 2 πn : x = 4 π + 6 πn

x = \frac{π}{2} + 6 πn, x = \frac{11 π}{2} + 6 πn, x = \frac{5 π}{2} + 6 πn, x = \frac{7 π}{2} + 6 πn, x = π + 6 πn, x = 5 π + 6 πn, x = 2 π + 6 πn, x = 4 π + 6 πn

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