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Answers
Answer:
Identity Used:
⇒ Sin²A + Cos²A = 1
According to the question, it is given that:
⇒ 7 Sin²A + 3 Cos²A = 4
Splitting 7 Sin²A as two components we get:
⇒ 4 Sin²A + 3 Sin²A + 3 Cos²A = 4
Taking 3 in common we get:
⇒ 4 Sin²A + 3 ( Sin²A + Cos²A ) = 4
Using the identity, we get:
⇒ 4 Sin²A + 3 ( 1 ) = 4
⇒ 4 Sin²A + 3 = 4
⇒ 4 Sin²A = 4 - 3 = 1
⇒ Sin²A = 1/4
⇒ Sin A = √ (1/4) = 1/2
Now we know that Sin 30° is 1/2.
Hence the value of angle A is 30 degrees.
Hence value of Tan A is:
⇒ Tan A = Tan 30° = 1/√3
Hence Proved.
♣ Qᴜᴇꜱᴛɪᴏɴ :
- If 7sin²θ + 3cos²θ = 4 , show that tanθ = 1/√3
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♣ ɢɪᴠᴇɴ :
- 7sin²θ + 3cos²θ = 4
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♣ ᴛᴏ ᴘʀᴏᴠᴇ :
- tanθ = 1/√3
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♣ ᴀɴꜱᴡᴇʀ :
We know sin²θ + cos²θ = 1
7sin²θ + 3cos²θ = 4
⇒ 7sin²θ + 3cos²θ = 4(1)
⇒ 7sin²θ + 3cos²θ = 4(sin²θ + cos²θ)
⇒ 7sin²θ + 3cos²θ = 4sin²θ + 4cos²θ
Take like terms :
⇒ 7sin²θ - 4sin²θ = 4cos²θ - 3cos²θ
⇒ 3sin²θ = 1cos²θ
⇒ 3sin²θ = cos²θ
Let's solve it further !!
3sin²θ = cos²θ
We know sinθ = Height/Hypotenuse and cosθ = Base/Hypotenuse
⇒ 3 × (Height/Hypotenuse)² = (Base/Hypotenuse)²
Apply rule : (a/b)² = a²/b²
⇒ 3 × (Height²/Hypotenuse²) = (Base²/Hypotenuse²)
Multiplying Both sides by Hypotenuse²
⇒ 3 × (Height²/Hypotenuse²) × Hypotenuse² = (Base²/Hypotenuse²) Hypotenuse²
⇒ 3 × Height² = Base²
Dividing both Sides by Base²
⇒ 3 × Height²/Base² = Base²/Base²
⇒ 3 × Height²/Base² = 1
Dividing both Sides by 3
⇒ (3 × Height²/Base²)/3 = 1/3
⇒ Height²/Base² = 1/3
⇒ (Height/Base)² = 1/3
We know tanθ = Height/Base
tan²θ = (Height/Base)²
⇒ tan²θ = 1/3
Take Square root on both sides
tan²θ = 1/3
⇒ √(tan²θ) = √(1/3)
⇒ tanθ = 1/√3
Hence Proved !!!!
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♣ ᴏᴛʜᴇʀ ᴛʀɪɢᴏɴᴏᴍᴇᴛʀɪᴄ ɪᴅᴇɴᴛɪᴛɪᴇꜱ :
sin(θ) = Height / Hypotenuse
cos(θ) = Base / Hypotenuse
tan(θ) = Height / Base
cosec(θ) = Hypotenuse / Height
sec(θ) = Hypotenuse / Base
tan(θ) = Base / Height
