if sin alpha is equal to 336 by 625 and 450 degrees is less than Alpha is less than 540 degrees then find the value of sin alpha divided by 4
Answers
Answer:
Step-by-step explanation:
=> Suppose sin α/4 = x, we get
cos α = 2 (1 - 2x²)² - 1
= 1 - 8x² + 8x⁴ ...(1)
=> According to the question, sin alpha is equal to 336 by 625
cos α = √1 - sin² α (∵ sin²α + cos²α = 1)
cos α =±√ 1 - (336/625)²
=> Here, we will take minus sign as α is in 2nd quadrant. thus,
cos α = -√(625+336)(625-336)/ (625)²
= - √961*289 / (625)²
= - 31*17 / 625
= - 527/625
=> By place this value in eq (1), we get
1 - 8x² + 8x⁴ = -527 / 625
x⁴ - x² + 144/625 = 0
x² = 1/2[1 ± √1 - 4*144 / 625]
x² = 1/2[1 ± √625-576 / 625]
x² = 1/2[1 ± √49 / 625]
= 1/2 (1±7/25)
x² = 1/2 (32/25) or x² = 1/2 (18/25)
x² = 16 / 25 or x² = 9/25
As, x = sin α/4 is positive for the specified values of α, we obtain
sin α/4 = 4/5 or sin α/4 = 3/5
=> Now, 450 degrees is less than Alpha is less than 540 degrees means the angle α/4 lies in the 2nd quadrant, with 112.5° < α/4 < 135°.
Thus, we should have
sin 135° < sinα/4 < 112.5°
=> sinα/4 > 1/√2
=> sinα/4 > 0.7
This eliminates the value sinα = 3/5 = 0.6
Hence, sinα/4 = 4/5.
Answer:
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