In a Δ A B C , a b = 2 + 3 , and ∠ C = 60 ° . Then the ordered pair ( ∠ A , ∠ B ) is equal to:

a b > 1 ⇒ ∠ A > ∠ B A + B = 120 ° ⇒ s i n ( 120 ° − B ) s i n B = 2 + 3 ⇒ 3 2 c o t B + 1 2 = 2 + 3 ⇒ c o t B = 2 3 3 2 + 3 ⇒ c o t B = 2 + 3 ⇒ B = 15 ° Hence, ( ∠ A , ∠ B ) = ( 105 ° , 15 ° )

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Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

MathematicsTrigonometric Ratios & IdentitiesMeasurement of AnglesMedium2 minPYQ_2015

MathematicsMediumsingle choice

In a $Δ A B C$ , $\frac{a}{b} = 2 + \sqrt{3}$ , and $∠ C = 60 ° .$ Then the ordered pair $(∠ A, ∠ B)$ is equal to:

Options:

Answer:

Solution:

$\begin{array}{l} \frac{a}{b} > 1 \Rightarrow ∠ A > ∠ B \end{array}$

$A + B = 120 °$

$\Rightarrow \frac{s i n (120 ° - B)}{s i n B} = 2 + \sqrt{3}$

$\begin{array}{l} ⇒ \frac{\sqrt{3}}{2} c o t B + \frac{1}{2} = 2 + \sqrt{3} \end{array}$

$\begin{array}{l} \Rightarrow c o t B = \frac{2}{\sqrt{3}} (\frac{3}{2} + \sqrt{3}) \end{array}$

$\Rightarrow c o t B = 2 + \sqrt{3}$

$\begin{array}{l} \Rightarrow B = 15 ° \end{array}$

Hence, $(∠ A, ∠ B) = (105 °, 15 °)$

Stream:JEESubject:MathematicsTopic:Trigonometric Ratios & IdentitiesSubtopic:Measurement of Angles

⏱ 2mℹ️ Source: PYQ_2015

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