Question

Найдите значения других трех основных тригонометрических функций,если 1 пример. sin a = √2/3 ; 0<a<π/2 2 пример. cos a = -√6/4 ; π/2<a<π Пожалуйста,немного разъясните - ничего в этом не понимаю (((

Answer 1

Есть основное тригонометрическое тождество
$\sin^2 \alpha +\cos^2 \alpha =1$
из него выражаем косинус
$\cos \alpha =\pm \sqrt{1-\sin^2 \alpha }$

в первом примере угол альфа в первой четверти, значит там косинус положительный, поэтому берем значение косинуса со знаком плюс
$> аналогично решаем второй пример. из основного тригонометрического тождества выражаем синус <img src=$
угол альфа во второй четверти, там синус положительный, значит берет синус со знаком плюс

$\sin \alpha = \sqrt{1-\cos^2 \alpha }= \sqrt{1- \dfrac{6}{16} } = \dfrac{ \sqrt{10}}{4} \\\\\tan \alpha = \dfrac{\sin \alpha }{\cos \alpha } = -\dfrac{ \sqrt{10}\cdot4 }{4\cdot \sqrt{6} } =- \dfrac{ \sqrt{10} }{ \sqrt{6} }$

Answer 2

Sine Function (sin)

The sine function (sin) is one of the three primary trigonometric functions. It relates the angle of a right triangle to the ratio of the length of the side opposite the angle to the length of the hypotenuse. The sine function is defined as:

sin(a) = opposite/hypotenuse

In the given example, sin(a) = √2/3, where 0 < a < π/2. This means that in a right triangle, the length of the side opposite angle a is √2, and the length of the hypotenuse is 3.

Cosine Function (cos)

The cosine function (cos) is another primary trigonometric function. It relates the angle of a right triangle to the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. The cosine function is defined as:

cos(a) = adjacent/hypotenuse

In the second example, cos(a) = -√6/4, where π/2 < a < π. This means that in a right triangle, the length of the side adjacent to angle a is -√6, and the length of the hypotenuse is 4.

Tangent Function (tan)

The tangent function (tan) is the third primary trigonometric function. It relates the angle of a right triangle to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. The tangent function is defined as:

tan(a) = opposite/adjacent

To find the value of the tangent function in the given examples, we can use the values of sin(a) and cos(a) that we already have. The tangent function can be calculated as:

tan(a) = sin(a)/cos(a)

Let's calculate the values of the tangent function for the given examples:

1. For the first example, sin(a) = √2/3 and cos(a) is not given. Therefore, we cannot determine the value of the tangent function without knowing the value of cos(a).

2. For the second example, cos(a) = -√6/4. Using this value, we can calculate the tangent function as:

tan(a) = sin(a)/cos(a) = (√2/3) / (-√6/4) = -2√2/√6 = -√2/√3

Therefore, the value of the tangent function for the second example is -√2/√3.

Please let me know if there's anything else I can help you with!