Answer to Question #350746 in Algebra for Biola

Question #350746

Find x if, 2^sin²x + 2^cos²x = 3


1
Expert's answer
2022-06-15T13:00:48-0400

"2^{\\sin^2x}+2^{\\cos^2x}=3"

"2^{\\sin^2x}+2^{1-\\sin^2x}=3"

"2^{\\sin^2x}+2\\cdot\\frac{1}{2^{\\sin^2x}}=3"


Let "y=2^{\\sin^2x}"


"y+\\frac{2}{y}=3"

"y^2-3y+2=0"

"y_1=1", "y_2=2"


"2^{\\sin^2x}=1"

"\\sin^2x=0"

"\\sin x=0"

"x=\\pi n, n\\in\\mathbb{Z}"


"2^{\\sin^2x}=2"

"\\sin^2x=1"

"\\sin x=\\pm1"

"x=(-1)^n\\frac{\\pi}{2}+\\pi n, n\\in\\mathbb{Z}"


Answer:

"x=\\pi n, n\\in\\mathbb{Z}"

"x=(-1)^n\\frac{\\pi}{2}+\\pi n, n\\in\\mathbb{Z}"





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