Question

Type the correct answer in numerals instead of words. If necessary, use / for the fraction bar. In right triangle ABC with right angle at vertex B, angle BAC measures 60 degrees. ��ABC is a right triangle in which ���B is a right angle, AB = 1, AC = 2, and BC = ?. cos C �� sin A =

Answer 1

BC is found using the Pythagorean theorem and equals √3, or approximately 1.732, in a right triangle with sides AB=1 and AC=2. Cosine of angle C (cos C) is 1/2 and sine of angle A (sin A) is √3/2.

Step-by-step explanation:

In the given right triangle ABC with a right angle at B, and angle BAC measuring 60 degrees, the side AB is given as 1, and the hypotenuse AC is given as 2. To find BC, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (AC) is equal to the sum of the squares of the other two sides (AB and BC). So, if AB = 1 and AC = 2, then the equation becomes:

AC² = AB² + BC²

2² = 1² + BC²

4 = 1 + BC²

BC² = 4 - 1

BC² = 3

BC = √3

Therefore, BC = √3, which is approximately 1.732.

For cos C and sin A, we use the definitions of cosine and sine in a right triangle. Cos C is the adjacent side over the hypotenuse, and since C is at vertex C, its adjacent side is AB and the hypotenuse is AC:

cos C = AB / AC = 1 / 2

sin A, which is the side opposite angle A over the hypotenuse, that is BC over AC:

sin A = BC / AC = √3 / 2

Thus, cos C = 1/2 and sin A = √3 / 2.