Question

В прямоугольном треугольнике АВС угол А = 40 градусов , угол С = 90 градусов , угол АВ = 2см . Найдите ВС, АС и угол В С решением пожалуйстааа

Answer 1

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Answer 2

Solution for the Triangle ABC

To find the lengths of sides AB and AC, as well as the angle B, we can use trigonometric ratios in a right-angled triangle.

Given information: - Angle A = 40 degrees - Angle C = 90 degrees - Side AB = 2 cm

To find side AC (hypotenuse) and side BC, we can use the trigonometric ratios sine, cosine, and tangent.

Let's start by finding side AC (hypotenuse): - We know that the sine of angle A is equal to the ratio of the opposite side (AB) to the hypotenuse (AC). - So, we can use the formula: sin(A) = AB/AC.

Now, let's find side BC: - We know that the cosine of angle A is equal to the ratio of the adjacent side (BC) to the hypotenuse (AC). - So, we can use the formula: cos(A) = BC/AC.

Finally, let's find angle B: - We know that the tangent of angle B is equal to the ratio of the opposite side (AB) to the adjacent side (BC). - So, we can use the formula: tan(B) = AB/BC.

Let's calculate the values:

1. Finding side AC (hypotenuse): - Using the formula sin(A) = AB/AC, we can rearrange it to find AC: AC = AB/sin(A). - Plugging in the values, we get AC = 2 cm / sin(40 degrees).

2. Finding side BC: - Using the formula cos(A) = BC/AC, we can rearrange it to find BC: BC = AC * cos(A). - Plugging in the values, we get BC = (2 cm / sin(40 degrees)) * cos(40 degrees).

3. Finding angle B: - Using the formula tan(B) = AB/BC, we can rearrange it to find B: B = arctan(AB/BC). - Plugging in the values, we get B = arctan(2 cm / [(2 cm / sin(40 degrees)) * cos(40 degrees)]).

Let's calculate the values using the given information:

1. Calculating side AC (hypotenuse): - AC = 2 cm / sin(40 degrees) = 3.08 cm.

2. Calculating side BC: - BC = (2 cm / sin(40 degrees)) * cos(40 degrees) = 2.35 cm.

3. Calculating angle B: - B = arctan(2 cm / [(2 cm / sin(40 degrees)) * cos(40 degrees)]) = 33.6 degrees.

Therefore, the values of side AC, side BC, and angle B are approximately: - Side AC = 3.08 cm - Side BC = 2.35 cm - Angle B = 33.6 degrees.

Please note that these values are approximate due to rounding.

I hope this helps! Let me know if you have any further questions.