Arctg(3) = a - это такой угол, что tg a = 3, a ∈ (-pi/2; pi/2). Тогда 1/cos^2 a = 1 + tg^2 a = 1 + 9 = 10 cos^2 a = 1/10; cos a = 1/√10 = √10/10 sin^2 a = 1 - cos^2 a = 1 - 1/10 = 9/10; sin a = 3/√10 = 3√10/10
arcsin(√5/5) = b - это такой угол, что sin b = √5/5, b ∈ (-pi/2; pi/2) sin^2 b = 1/5; cos^2 b = 1- sin^2 b = 4/5; cos b = 2/√5 = 2√5/5
Найдем sin(a - b) = sin(arctg(3) - arcsin(√5/5)) sin(a - b) = sin a *cos b - cos a*sin b = = 3√10/10*2√5/5 - √10/10*√5/5 = 6*√50/50 - √50/50 = 5*5√2/50 = √2/2 Если sin(a - b) = √2/2, то a - b = pi/4
1/cos^2 a = 1 + tg^2 a = 1 + 9 = 10
cos^2 a = 1/10; cos a = 1/√10 = √10/10
sin^2 a = 1 - cos^2 a = 1 - 1/10 = 9/10; sin a = 3/√10 = 3√10/10
arcsin(√5/5) = b - это такой угол, что sin b = √5/5, b ∈ (-pi/2; pi/2)
sin^2 b = 1/5; cos^2 b = 1- sin^2 b = 4/5; cos b = 2/√5 = 2√5/5
Найдем sin(a - b) = sin(arctg(3) - arcsin(√5/5))
sin(a - b) = sin a *cos b - cos a*sin b =
= 3√10/10*2√5/5 - √10/10*√5/5 = 6*√50/50 - √50/50 = 5*5√2/50 = √2/2
Если sin(a - b) = √2/2, то a - b = pi/4