Trigonometry
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What is the value of (cotθ + cosecθ - 1) ? (cotθ - cosecθ + 1)
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Expression = (cotθ + cosecθ - 1) (cotθ - cosecθ + 1) Expression = cotθ + cosecθ - (cosec2θ - cot2θ) (cotθ - cosecθ + 1) Expression = (cotθ + cosecθ) - (cosecθ - cotθ)(cosecθ + cotθ) (cotθ - cosecθ + 1) Expression = (cotθ + cosecθ)(1 - cosecθ + cotθ) = cotθ + cosecθ (cotθ - cosecθ + 1) Correct Option: A
Expression = (cotθ + cosecθ - 1) (cotθ - cosecθ + 1) Expression = cotθ + cosecθ - (cosec2θ - cot2θ) (cotθ - cosecθ + 1) Expression = (cotθ + cosecθ) - (cosecθ - cotθ)(cosecθ + cotθ) (cotθ - cosecθ + 1) Expression = (cotθ + cosecθ)(1 - cosecθ + cotθ) = cotθ + cosecθ (cotθ - cosecθ + 1)
- Find the value of 8 cos 10° cos20° cos 40°.
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Expression = 8 cos 10°. cos 20° . cos 40°
Expression = 4 2 sin 10° .cos 10°. cos 20° . cos 40° sin 10° Expression = 2 2 sin 20° . cos 20° . cos 40° sin 10°
{ ∴ 2 sinθ . cosθ = sin2θ }Expression = 2 sin 40° . cos 40° sin 10° Expression = sin 80° = sin 80° sin 10° cos (90° - 10°) Expression = sin 80° or cos 10° cos 80° sin 10°
Expression = tan80° or cot 10°
{ ∴ cos (90° - θ) = sinθ and sin (90° - θ) = cosθ }Correct Option: C
Expression = 8 cos 10°. cos 20° . cos 40°
Expression = 4 2 sin 10° .cos 10°. cos 20° . cos 40° sin 10° Expression = 2 2 sin 20° . cos 20° . cos 40° sin 10°
{ ∴ 2 sinθ . cosθ = sin2θ }Expression = 2 sin 40° . cos 40° sin 10° Expression = sin 80° = sin 80° sin 10° cos (90° - 10°) Expression = sin 80° or cos 10° cos 80° sin 10°
Expression = tan80° or cot 10°
{ ∴ cos (90° - θ) = sinθ and sin (90° - θ) = cosθ }
- If secθ + tanθ = 2, then the value of sinθ is :
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secθ + tanθ = 2
∴ sec2θ – tan2θ = 1
⇒ (secθ + tanθ)(secθ – tanθ) = 1⇒ secθ – tanθ = 1 2 ∴ secθ + tanθ + secθ – tanθ = 2 + 1 2 ⇒ 2secθ = 5 ⇒ secθ = 5 2 4 Again, (secθ + tanθ) – (secθ – tanθ) = 2 - 1 2 ⇒ 2tanθ = 3 ⇒ tanθ = 3 2 4 ⇒ sinθ = tanθ = 3 ÷ 5 = 3 secθ 4 4 5 Correct Option: D
secθ + tanθ = 2
∴ sec2θ – tan2θ = 1
⇒ (secθ + tanθ)(secθ – tanθ) = 1⇒ secθ – tanθ = 1 2 ∴ secθ + tanθ + secθ – tanθ = 2 + 1 2 ⇒ 2secθ = 5 ⇒ secθ = 5 2 4 Again, (secθ + tanθ) – (secθ – tanθ) = 2 - 1 2 ⇒ 2tanθ = 3 ⇒ tanθ = 3 2 4 ⇒ sinθ = tanθ = 3 ÷ 5 = 3 secθ 4 4 5
- ∠Y is the right angle of the trianlge XYZ. If XY = 2 √6 cm and XZ – YZ = 2cm, then the value of (secX + tanX) is :
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XY = 2 6 cm
XY – YZ = 2 cm. ...(i)
∴ XZ2 = XY2 + YZ2
⇒ XZ2 - YZ2 = (2√6)2
⇒ XZ2 - YZ2 = 24∴ XZ2 - YZ2 = 24 XZ - YZ 2
⇒ XZ + YZ = 12 ....(ii)⇒ secX + tan X = XZ + YZ XY XY ⇒ secX + tan X = XZ + YZ = XY ⇒ secX + tan X = XZ + YZ = 12 = √6 XY 2√6
Correct Option: D
XY = 2 6 cm
XY – YZ = 2 cm. ...(i)
∴ XZ2 = XY2 + YZ2
⇒ XZ2 - YZ2 = (2√6)2
⇒ XZ2 - YZ2 = 24∴ XZ2 - YZ2 = 24 XZ - YZ 2
⇒ XZ + YZ = 12 ....(ii)⇒ secX + tan X = XZ + YZ XY XY ⇒ secX + tan X = XZ + YZ = XY ⇒ secX + tan X = XZ + YZ = 12 = √6 XY 2√6
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If α + θ = 7π and tanθ = √3 , then the value of tanθ is : 12
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tanθ = √3 = tan π 3 ⇒ θ = π 3 ∴ α + θ = 7π 12 ⇒ α = 7π - π 12 3 ⇒ α = 7π - π = π 12 4 ∴ tanα = tan π = 1 4
Correct Option: B
tanθ = √3 = tan π 3 ⇒ θ = π 3 ∴ α + θ = 7π 12 ⇒ α = 7π - π 12 3 ⇒ α = 7π - π = π 12 4 ∴ tanα = tan π = 1 4