Question

Дан прямоугольный треугольник АВС, у которого угол С прямой, ВС=6 см, угол А=60. Найти АВ и АС.

Answer 1

SinCAB = CB/AB;
sin60=CB/AB
√3/2=6/AB;
AB=4√3
AC²=AB²-CB² 
AC²=(4√3)²-6²=√12=2√3
сторона АВ = 4√3
сторона АС = 2√3

Answer 2

Given Information:

We are given a right triangle ABC, where angle C is a right angle, BC = 6 cm, and angle A is 60 degrees. We need to find the lengths of AB and AC.

Solution:

To find the lengths of AB and AC, we can use trigonometric ratios in a right triangle.

Let's start by finding the length of AB.

Using the trigonometric ratio for the sine of angle A, we have:

sin(A) = opposite/hypotenuse

In this case, the opposite side of angle A is AB, and the hypotenuse is AC.

Therefore, we can write:

sin(60) = AB/AC

To find AB, we need to know the length of AC. Let's find the length of AC next.

Using the Pythagorean theorem, we have:

AC^2 = AB^2 + BC^2

Substituting the given values, we have:

AC^2 = AB^2 + 6^2

Now, let's substitute the value of AC^2 in the equation for sine:

sin(60) = AB/sqrt(AB^2 + 6^2)

Simplifying the equation, we have:

sqrt(3)/2 = AB/sqrt(AB^2 + 36)

Cross-multiplying, we get:

sqrt(3) * sqrt(AB^2 + 36) = 2 * AB

Squaring both sides of the equation, we have:

3 * (AB^2 + 36) = 4 * AB^2

Expanding and simplifying, we get:

3AB^2 + 108 = 4AB^2

Rearranging the equation, we have:

AB^2 = 108

Taking the square root of both sides, we have:

AB = sqrt(108)

Simplifying, we get:

AB = 6 * sqrt(3)

Now, let's substitute the value of AB in the equation for AC^2:

AC^2 = (6 * sqrt(3))^2 + 6^2

Simplifying, we have:

AC^2 = 108 + 36

AC^2 = 144

Taking the square root of both sides, we have:

AC = 12

Therefore, the length of AB is 6 * sqrt(3) cm and the length of AC is 12 cm.

Answer:

The length of AB is 6 * sqrt(3) cm and the length of AC is 12 cm.