If a cos- -b sin -4 and a sin-b cos-3- then what is the value of a2-b2-

A cosθ + b sin θ=4 and a sinθ b cosθ=3 Squaring both the equations, (a cosθ nbsp;+ b sinθ)^2 = 4^2 and (a sinθ b cosθ)^2 = 3^2 Adding, we get a^2c ...

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Mathematics

Trigonometric Ratios

If a cosθ +b ...

Question

If $a c o s θ + b s i n θ = 4$ and $a s i n θ - b c o s θ = 3$ , then what is the value of $a^{2} + b^{2}$ ?
25

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Solution

The correct option is A 25
$a c o s θ + b s i n θ = 4$ and $a s i n θ - b c o s θ = 3$
Squaring both the equations,
$(a c o s θ + b s i n θ)^{2} = 4^{2} and (a s i n θ - b c o s θ)^{2} = 3^{2} Adding, we get a^{2} c o s^{2} θ + 2 a b s i n θ c o s θ + b^{2} s i n^{2} θ + a^{2} s i n^{2} θ - 2 a b s i n θ c o s θ + b^{2} c o s^{2} θ = 16 + 9 \Rightarrow a^{2} (c o s^{2} θ + s i n^{2} θ) + b^{2} (c o s^{2} θ + s i n^{2} θ) = 25 \Rightarrow a^{2} + b^{2} = 25$

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If a cosθ+b sinθ=4 and a sinθ−b cosθ=3, then what is the value of a2+b2?25

The correct option is A 25a cosθ+b sinθ=4 and a sinθ−b cosθ=3 Squaring both the equations, (a cosθ+b sinθ)2=42and (a sinθ−b cosθ)2=32Adding, we geta2cos2θ+2ab sinθcosθ+b2sin2θ+a2sin2θ−2ab sinθcosθ+b2cos2θ=16+9⇒a2(cos2θ+sin2θ)+b2(cos2θ+sin2θ)=25⇒a2+b2=25

If $a c o s θ + b s i n θ = 4$ and $a s i n θ - b c o s θ = 3$ , then what is the value of $a^{2} + b^{2}$ ?
25