NDA 2019
Exam NDA
Year 2019
Subject Mathematics
Chapter Trigonometric Ratios & Identities NDA 2019 · Mathematics · Trigonometric Ratios & Identities
What is the value of 34^ ^ 236^ ^ - 56^ ^ 124^ ^ 28^ ^ 88^ ^ + 178^...
What is the value of
\frac{\sin 34^{{}^{\circ}} \cos 236^{{}^{\circ}}-\sin 56^{{}^{\circ}} \sin 124^{{}^{\circ}}}{\cos 28^{{}^{\circ}} \cos 88^{{}^{\circ}}+\cos 178^{{}^{\circ}} \sin 208^{{}^{\circ}}} ?
Step-by-step solution
sin 3 4 ∘ cos 23 6 ∘ − sin 5 6 ∘ sin 12 4 ∘ cos 2 8 ∘ cos 8 8 ∘ + cos 17 8 ∘ sin 20 8 ∘ \frac{\sin 34^{{}^{\circ}} \cos 236^{{}^{\circ}}-\sin 56^{{}^{\circ}} \sin 124^{{}^{\circ}}}{\cos 28^{{}^{\circ}} \cos 88^{{}^{\circ}}+\cos 178^{{}^{\circ}} \sin 208^{{}^{\circ}}} c o s 2 8 ∘ c o s 8 8 ∘ + c o s 17 8 ∘ s i n 20 8 ∘ s i n 3 4 ∘ c o s 23 6 ∘ − s i n 5 6 ∘ s i n 12 4 ∘ = sin 3 4 ∘ ( − cos 5 6 ∘ ) − sin 5 6 ∘ cos 3 4 ∘ cos 2 8 ∘ sin 2 ∘ + cos 2 ∘ sin 2 8 ∘ = − sin 3 4 ∘ cos 5 6 ∘ − sin 5 6 ∘ cos 3 4 ∘ sin ( 2 8 ∘ + 2 ∘ ) \begin{aligned} & =\frac{\sin 34^{{}^{\circ}}\left(-\cos 56^{{}^{\circ}}\right)-\sin 56^{{}^{\circ}} \cos 34^{{}^{\circ}}}{\cos 28^{{}^{\circ}} \sin 2^{{}^{\circ}}+\cos 2^{{}^{\circ}} \sin 28^{{}^{\circ}}} \\ & =\frac{-\sin 34^{{}^{\circ}} \cos 56^{{}^{\circ}}-\sin 56^{{}^{\circ}} \cos 34^{{}^{\circ}}}{\sin \left(28^{{}^{\circ}}+2^{{}^{\circ}}\right)}\end{aligned} = cos 2 8 ∘ sin 2 ∘ + cos 2 ∘ sin 2 8 ∘ sin 3 4 ∘ ( − cos 5 6 ∘ ) − sin 5 6 ∘ cos 3 4 ∘ = sin ( 2 8 ∘ + 2 ∘ ) − sin 3 4 ∘ cos 5 6 ∘ − sin 5 6 ∘ cos 3 4 ∘ = − sin ( 3 4 ∘ + 5 6 ∘ ) sin 3 0 ∘ = − sin 9 0 ∘ sin 3 0 ∘ = − 1 1 2 = − 2 =-\frac{\sin \left(34^{{}^{\circ}}+56^{{}^{\circ}}\right)}{\sin 30^{{}^{\circ}}}=\frac{-\sin 90^{{}^{\circ}}}{\sin 30^{{}^{\circ}}}=\frac{-1}{\frac{1}{2}}=-2 = − s i n 3 0 ∘ s i n ( 3 4 ∘ + 5 6 ∘ ) = s i n 3 0 ∘ − s i n 9 0 ∘ = 2 1 − 1 = − 2
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