Worksheet: Solving Trigonometric Equations with the Double-Angle Identity

In this worksheet, we will practice solving trigonometric equations using the double-angle identity.

Q1:

If 0β‰€πœƒ<180∘∘, find the solution set of √2πœƒπœƒβˆ’πœƒ=0sincossin.

  • A{45,90}∘∘
  • B{0,135}∘∘
  • C{45,135}∘∘
  • D{0,45}∘∘

Q2:

Find πœƒ in degrees given sincosπœƒ=4πœƒ where πœƒ is a positive acute angle.

Q3:

Find the general solution to the equation cossin3π‘₯=π‘₯4.

  • Aπ‘₯=2πœ‹13+2π‘›πœ‹13, π‘₯=βˆ’2πœ‹11+2π‘›πœ‹11, where π‘›βˆˆβ„€
  • Bπ‘₯=πœ‹12+π‘›πœ‹3, π‘₯=βˆ’2πœ‹11+8π‘›πœ‹11, where π‘›βˆˆβ„€
  • Cπ‘₯=2πœ‹13+4π‘›πœ‹13, π‘₯=βˆ’2πœ‹11+4π‘›πœ‹11, where π‘›βˆˆβ„€
  • Dπ‘₯=πœ‹12+π‘›πœ‹3, π‘₯=2πœ‹13+8π‘›πœ‹13, where π‘›βˆˆβ„€
  • Eπ‘₯=2πœ‹13+8π‘›πœ‹13, π‘₯=βˆ’2πœ‹11+8π‘›πœ‹11, where π‘›βˆˆβ„€

Q4:

Find the set of possible solutions of 2πœƒπœƒ=0sincos given πœƒβˆˆ[0,360)∘∘.

  • A{30,150,180,270}∘∘∘∘
  • B{60,120,180,270}∘∘∘∘
  • C{0,90,180,270}∘∘∘∘
  • D{0,90,120,240}∘∘∘∘

Q5:

Find the solution set for π‘₯ given coscos2π‘₯+13√3π‘₯=βˆ’19 where π‘₯∈(0,2πœ‹).

  • A{150,210}∘∘
  • B{120,240}∘∘
  • C{150,330}∘∘
  • D{30,330}∘∘

Q6:

Find all the possible solutions, that is, the general solution, of the equation sincossinπœƒπœƒ=√22πœƒ.

  • AΒ±πœ‹4+2π‘›πœ‹ (where π‘›βˆˆβ„€)
  • Bπ‘›πœ‹, βˆ’πœ‹2+2π‘›πœ‹ (where π‘›βˆˆβ„€)
  • Cπ‘›πœ‹, πœ‹4+2π‘›πœ‹ (where π‘›βˆˆβ„€)
  • Dπ‘›πœ‹, Β±πœ‹4+2π‘›πœ‹ (where π‘›βˆˆβ„€)
  • Eπ‘›πœ‹, Β±πœ‹2+2π‘›πœ‹ (where π‘›βˆˆβ„€)

Q7:

If 0β‰€πœƒ<180∘∘, find the solution set of 2πœƒπœƒ+πœƒ=0sincossin.

  • A{0,30}∘∘
  • B{90,120}∘∘
  • C{0,60}∘∘
  • D{0,120}∘∘

Q8:

Find the solution set for π‘₯ given coscos2π‘₯+5√3π‘₯=βˆ’7 where π‘₯∈(0,2πœ‹).

  • A{30,330}∘∘
  • B{150,210}∘∘
  • C{60,240}∘∘
  • D{30,300}∘∘

Q9:

Solve tansinο€»π‘₯2=π‘₯, where 0≀π‘₯<2πœ‹.

  • Aπ‘₯∈0,πœ‹2
  • Bπ‘₯∈0,πœ‹2,3πœ‹2,2πœ‹οΈ
  • Cπ‘₯∈0,πœ‹4,πœ‹,3πœ‹4
  • Dπ‘₯∈0,πœ‹2,3πœ‹2
  • Eπ‘₯∈0,πœ‹4

Q10:

By using the half angle formula sincosο€»π‘₯2=ο„ž1βˆ’π‘₯2, or otherwise, solve the equation sincosο€»π‘₯2+π‘₯=1, where 0≀π‘₯<2πœ‹.

  • Aπ‘₯=0,13πœ‹,53πœ‹οΈ
  • Bπ‘₯=0,16πœ‹,56πœ‹οΈ
  • Cπ‘₯=0,13πœ‹,56πœ‹οΈ
  • Dπ‘₯=0,13πœ‹οΈ
  • Eπ‘₯=0,16πœ‹οΈ

Q11:

Find the solution set for π‘₯ given coscossinsinπ‘₯2π‘₯βˆ’π‘₯2π‘₯=12 where 0<π‘₯<360∘∘.

  • A{20,110}∘∘
  • B{10,110}∘∘
  • C{10,100}∘∘
  • D{20,100}∘∘

Q12:

Find all the possible general solutions of cossincosπœƒπœƒ=√22πœƒ.

  • A2π‘›πœ‹+πœ‹2, πœ‹4+2π‘›πœ‹, βˆ’πœ‹4+πœ‹+2π‘›πœ‹
  • B2π‘›πœ‹Β±πœ‹2, πœ‹4+2π‘›πœ‹, βˆ’πœ‹4+πœ‹+2π‘›πœ‹
  • C2π‘›πœ‹Β±πœ‹2, πœ‹4+2π‘›πœ‹, βˆ’πœ‹4+πœ‹
  • D2π‘›πœ‹Β±πœ‹2, πœ‹4+2π‘›πœ‹, πœ‹4+πœ‹+2π‘›πœ‹
  • E2π‘›πœ‹βˆ’πœ‹2, πœ‹4+2π‘›πœ‹, βˆ’πœ‹4+πœ‹+2π‘›πœ‹

Q13:

Find the set of values satisfying cos2π‘₯=βˆ’βˆš32, where 0≀π‘₯<2πœ‹.

  • A5πœ‹6,7πœ‹6,11πœ‹6
  • B{0,πœ‹}
  • C5πœ‹12,7πœ‹12
  • D5πœ‹12,7πœ‹12,17πœ‹12,19πœ‹12
  • E5πœ‹6,7πœ‹6

Q14:

Find the value of π‘₯ given cossin2π‘₯=3π‘₯ where π‘₯ is an acute angle. Give the answer to the nearest degree.

Q15:

Find the value of 𝑋 without using a calculator, given 𝑋7πœ‹6οˆο€»πœ‹3=ο€Ό2πœ‹3οˆο€Ό5πœ‹6sincostansin.

  • Aβˆ’12
  • Bβˆ’112
  • C112
  • D12

Q16:

Find the general solution to the equation sincos2π‘₯=π‘₯2.

  • Aπ‘₯=πœ‹2+2πœ‹π‘›, π‘₯=πœ‹5+4π‘›πœ‹5, where π‘›βˆˆβ„€
  • Bπ‘₯=πœ‹2+2πœ‹π‘›, π‘₯=πœ‹3+4π‘›πœ‹3, where π‘›βˆˆβ„€
  • Cπ‘₯=πœ‹2+2πœ‹π‘›, π‘₯=πœ‹+2πœ‹π‘›, where π‘›βˆˆβ„€
  • Dπ‘₯=πœ‹+2πœ‹π‘›, π‘₯=πœ‹3+4π‘›πœ‹3, where π‘›βˆˆβ„€
  • Eπ‘₯=πœ‹5+4π‘›πœ‹5, π‘₯=πœ‹3+4π‘›πœ‹3, where π‘›βˆˆβ„€

Q17:

Find the value of 𝑋 given cos2𝑋=√32 where 2𝑋 is an acute angle. Give the answer to the nearest minute.

  • A2230β€²βˆ˜
  • B30∘
  • C15∘
  • D45∘

Q18:

Find the value of 𝑋 in degrees given cossinsintansin3𝑋=30604545∘∘∘∘ where 3𝑋 is an acute angle.

Q19:

Find the value of πœƒ given sinsincotcossinπœƒ=120780+240750∘∘∘∘ giving the answer to the nearest second.

  • Aπœƒ=34531β€²21β€²β€²βˆ˜ or πœƒ=19428β€²39β€²β€²βˆ˜
  • Bπœƒ=1428β€²39β€²β€²βˆ˜ or πœƒ=16531β€²21β€²β€²βˆ˜
  • Cπœƒ=19428β€²39β€²β€²βˆ˜ or πœƒ=16531β€²21β€²β€²βˆ˜
  • Dπœƒ=1428β€²39β€²β€²βˆ˜ or πœƒ=19428β€²39β€²β€²βˆ˜

Q20:

Find the set of possible values of π‘₯ which satisfy 1√π‘₯βˆ’π‘₯=2coscosοŠͺ where 0<π‘₯<360∘∘.

  • A{45,150,240,300}∘∘∘∘
  • B{45,135,225,315}∘∘∘∘
  • C{45,135}∘∘
  • D{45,135,210,330}∘∘∘∘

Q21:

Find π‘šβˆ πœƒ given cossinsincos34.534.5+1269=πœƒβˆ˜βˆ˜βˆ˜ where πœƒ is a positive acute angle.

Q22:

Find the set of solutions for π‘₯ given sincoscossin9π‘₯4π‘₯βˆ’9π‘₯4π‘₯=√22 where 0<π‘₯<2πœ‹5∘ .

  • A{9,30}∘∘
  • B{9,27}∘∘
  • C{6,27}∘∘
  • D{6,30}∘∘

Q23:

Find the set of solutions in the range 0<π‘₯<180 for the equation (π‘₯+π‘₯)=22π‘₯sincossin.

  • A{45,105,165}∘∘∘
  • B{45,75,105}∘∘∘
  • C{15,75,90}∘∘∘
  • D{90,210,330}∘∘∘
  • E{45,75,165}∘∘∘

Q24:

Given that sincos𝑋+𝑋=βˆ’713 and πœ‹<𝑋<3πœ‹2, determine the possible values of cos2𝑋.

  • A120169,βˆ’120169
  • B169119,βˆ’169119
  • C119169,βˆ’119169
  • D169120,βˆ’169120

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