Question

$2\pi -[sin^{-1} \frac{4}{5} +sin^{-1} \frac{5}{13} +sin^{-1} \frac{16}{65}] ?$

Answer 1

Hlo Strong Girl ✨ ur answer is in attachment

Answer 2

we have to find the value 2π - [sin¯¹(4/5) + sin¯¹(5/13) + sin¯¹(16/65) ]

solution : first solve sin¯¹(4/5) + sin¯¹(5/13)

we know, sin¯¹A + sin¯¹B = sin¯¹[A√(1 - B²) + B√(1 - A²)]

so, sin¯¹(4/5) + sin¯¹(5/13) = sin¯¹[4/5√(1 - (5/13)²) + 5/13√(1 - (4/5)²]

= sin¯¹[4/5 × 12/13 + 5/13 × 3/5 ]

= sin¯¹[48/65 + 15/65]

= sin¯¹ (63/65)

let sin¯¹(63/65) = A ⇒sinA = 63/65

so, cosA = √(65² - 63²)/65 = 16/65

⇒A = cos¯¹(16/65)

⇒sin¯¹(63/65) = cos¯¹(16/65)

hence, sin¯¹(4/5) + sin¯¹(5/13) = cos¯¹(16/65)

now 2π - [cos¯¹(16/65) + sin¯¹(16/65)]

we know, sin¯¹x + cos¯¹x = π/2

so, cos¯¹(16/65) + sin¯¹(16/65) = π/2

2π - [cos¯¹(16/65) + sin¯¹(16/65)] = 2π - π/2 = 3π/2

Therefore the value of 2π - [cos¯¹(16/65) + sin¯¹(16/65)] is 3π/2