Question

If x=2sin2Ɵ and y=2cos2Ɵ+1, then find x+y​

Answer 1

Given:-

• x = 2 sin²θ
• y = 2 cos²θ + 1

⠀

To find:-

• x + y

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Solution:-

According to the question

→ 2 sin²θ + 2 cos²θ + 1

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→ 2 (sin²θ + cos²θ) + 1

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• Putting sin²θ + cos²θ = 1

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→ 2 × 1 + 1

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→ 2 + 1

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→ 3

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Hence,

• x + y = 3

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⛦Trigonometric Table:-

$\bullet\:\sf Trigonometric\:Values :\\\\\boxed{\begin{tabular}{c|c|c|c|c|c}Radians/Angle & 0 & 30 & 45 & 60 & 90\\\cline{1-6}Sin \theta & 0 & \dfrac{1}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{\sqrt{3}}{2} & 1\\\cline{1-6}Cos \theta & 1 & \dfrac{\sqrt{3}}{2}& \dfrac{1}{\sqrt{2}}& \dfrac{1}{2}&0\\\cline{1-6}Tan \theta&0& \dfrac{1}{\sqrt{3}}&1&\sqrt{3}&Not D{e}fined\end{tabular}}$

Answer 2

Given :

$\: \:$

• x = 2sin²θ
• y = 2cos²θ + 1

$\:$

To Find :

$\:$

• Value of x + y

⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀_______________

$\:$

Put the values of x and y in the equation.

$\: \:$

$\sf:\implies \: x + y = 2 {sin}^{2} \theta + 2 {cos}^{2} \theta + 1 \\ \\ \\ \sf:\implies \: x + y = 2( {sin}^{2} \theta + cos {}^{2} \theta) + 1 \\ \\ \\ \sf:\implies \: x + y = 2(1) + 1 \\ \\ \\ \sf as \: we\:know \: that \left( {sin}^{2}\theta + {cos}^{2} \theta = 1 \right) \\ \\ \\ \sf:\implies \: x + y = 2 + 1 \\ \\ \\ \sf:\implies \: x + y = \underline{\boxed{\sf\purple{3}}}\bigstar \\ \\ \\ \\ \sf\therefore{\underline{Hence ,\: the \: value \: of \:( x \: + y) \: is \:\bold{3} }}$

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More to know :

$\:$

• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ