Trigo Problem Sheet-1

Trigo Problem Sheet-1 - UPCSE Maths Course Week 0 Problem...

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UPCSE Maths Course Problem Sheet 0 Week: 0 (based on Lecture 0) Topic: Trigonometry (Self Study). 1. Prove that (cot θ + cosec θ ) 2 = 1 + cos θ 1 - cos θ . 2. Use the formula for cos( A + B ) to show that cos 2 θ = 1 2 (cos 2 θ + 1) . 3. Express 2 cos 2 θ - 3 sin 2 θ in the form R cos(2 θ + k ) , where 0 < k < π/ 2 . Hence obtain the general solution to 2 cos 2 θ - 3 sin 2 θ = 13 2 . 4. Two acute angles α and β are such that tan α = 1 2 and tan( α + β ) = 2 . Without finding α and β , ( i ) show that tan β = 3 4 ; ( ii ) evaluate sin α and sin β ; ( iii ) evaluate sin 2 2 α and sin 2 2 β . 5. Show that, for x ( - π, π ) , tan - 1 ( x ) - tan - 1 1 x = tan - 1 x 2 - 1 2 x . 6. Prove the following trigo identity: cos 2 θ = cosec θ - sin θ cosec θ . 7. Solve the trigo equation cos 5 θ - cos θ = sin 3 θ ; 0 θ 2 π . 8. Given that sec A = cos B + sin B , show that tan 2 A = sin 2 B . 9. Find x , such that sin - 1 x + tan - 1 x = π/ 2 . 10. Eliminate θ OR If x = a + r cos θ , and y = b + r sin θ , prove that x 2 + y 2 - 2 ax - 2 by + ( a 2 + b 2 - r 2 ) = 0 . 11. Prove that cos 120 + 3 tan 30 = sin 30 . 12. Prove that tan 4 θ = 4 tan θ - 4 tan 3 θ 1 - 6 tan 2 θ + tan 4 θ . 13. Solve for θ , sin ( θ + 30 ) = cos ( θ + 60 ) .
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