jika f(x)=sinx dan g(2x-3/4 phi)=akar dari (1-(f'(x))^2), maka nilai g'(5/12 phi) = ....

Fungsi turunan trigonometri

f(x)= sin x
f '(x)= cos x
g(2x - ³/₄ π)= √{1 - (f'(x))²} = √(1 - cos² x) = √(sin² x) = sin x

nilai g(⁵/₁₂ π) = ....

2x - ³/₄ π = ⁵/₁₂ π ....kalikan 12
24x - 9π = 5π
24x = 14π
x = 14π/24
x = ⁷/₁₂ π = 7/12 x 180° = 105

g(⁵/₁₂ π) = sin (⁷/₁₂ π)= sin 105°

sin 105 = sin (60+45) = sin 60 cos 45 + cos 60 sin 45
sin 105 = (1/2 √3)(1/2 √2) + (1/2)(1/2√2)
sin 105 = 1/4√6 + 1/4√2   = 1/4(√6 + √2)

g(⁵/₁₂ π) = 1/4(√6+√2)




....

f(x) = sin x => f'(x) = cos x
g(2x - 3π/4) = √(1 - (f'(x))^2)
g(2x - 3π/4) = √(1 - cos^2 x)
g(2x - 3π/4) = √(sin^2 x)
g(2x - 3π/4) = sin x
Misal
2x - 3π/4 = a => 2x = a + 3π/4 => x = 1/2 a + 3π/8
g(2x - 3π/4) = sin x
g(a) = sin (1/2 a + 3π/8)
g(x) = sin (1/2 x + 3π/8)
g'(x) = cos (1/2 x + 3π/8) . 1/2
g'(x) = 1/2 cos (1/2 x + 3π/8)
g'(5π/12) = 1/2 cos (1/2 . 5π/12 + 3π/8)
= 1/2 cos (5π/24 + 9π/24)
= 1/2 cos (14π/24)
= 1/2 cos 7π/12
= 1/2 cos 105°
= 1/2 cos (60° + 45°)
= 1/2 [cos 60° cos 45° - sin 60° sin 45°]
= 1/2 [1/2 . 1/2 √2 - 1/2 √3 . 1/2 √2]
= 1/2 [1/4 √2 - 1/4 √6]
= 1/8 √2 - 1/8 √6

= 1/8 (√2 - √6)

= 1/8 √2 (1 - √3)