Question

Приближенное значение sin(pi/6+0,03) cos(pi/6+0,04)

Answer 1

Ответ:

Объяснение:

$f(x+x_{0} )$≈$f(x)+f^{'} (x)x_{0}$

1) f(x)=sinx; x=π/6; x₀=0,03

$f^{'} (x)=cosx$

sin(π/6+0,03)≈sinπ/6+0,03·cosπ/6=0,5+0,03·√3/2=0,5+3·√3/200=

=(100+3·√3)/200

2) f(x)=cosx; x=π/6; x₀=0,04

$f^{'} (x)=-sinx$

cos(π/6+0,04)≈cosπ/6+0,04·(-sinπ/6)=√3/2+0,5·0,03=√3/2+0,015

Answer 2

To calculate the approximate values of sin(pi/6 + 0.03) and cos(pi/6 + 0.04), you can use the small-angle approximations for sine and cosine functions. These approximations work well when the angle is close to zero.

Small-angle approximations:

1. sin(x) ≈ x (for small x in radians)
2. cos(x) ≈ 1 - (x^2)/2 (for small x in radians)

Let's calculate the approximate values step by step:

1. For sin(pi/6 + 0.03): Using the small-angle approximation sin(x) ≈ x: sin(pi/6 + 0.03) ≈ pi/6 + 0.03

   Now, substitute the value of pi/6 (approximately 0.5236): sin(pi/6 + 0.03) ≈ 0.5236 + 0.03

   Calculate the result: sin(pi/6 + 0.03) ≈ 0.5536

2. For cos(pi/6 + 0.04): Using the small-angle approximation cos(x) ≈ 1 - (x^2)/2: cos(pi/6 + 0.04) ≈ 1 - ((pi/6 + 0.04)^2)/2

   Now, substitute the value of pi/6 (approximately 0.5236): cos(pi/6 + 0.04) ≈ 1 - ((0.5236 + 0.04)^2)/2

   Calculate the result: cos(pi/6 + 0.04) ≈ 1 - (0.5636^2)/2 ≈ 1 - 0.31698256 ≈ 0.68301744

Approximate values:

1. sin(pi/6 + 0.03) ≈ 0.5536
2. cos(pi/6 + 0.04) ≈ 0.6830

Please note that these are approximate values due to the use of small-angle approximations. For more precise values, you would need to use a calculator or a computer program that handles trigonometric functions accurately.