Given the indicated parts of triangle ABC with γ = 90°, approximate the remaining parts. (Round your answers for a and c to one decimal place.)

β = 51°31', b = 250.0
α = ° '
a =
c =

Given the indicated parts of triangle ABC with γ = 90°, approximate the remaining parts. (Round your answers for b and c to one decimal place.)
α = 27°50', a = 530
β = ° '
b =
c =

Given the indicated parts of triangle ABC with γ = 90°, express the third part in terms of the first two.
β, c; b
b =

Given the indicated parts of triangle ABC with γ = 90°, express the third part in terms of the first two.
α, a; b
b =

In every case, recall that if a = AB and b = BC and c = AC

α + β = 90°
sin α = cos β = b/c
cos α = sin β = a/c
tan α = cot β = b/a

Given the indicated parts of triangle ABC with 𝛾 = 90°, approximate the remaining parts. (Round your answers to the nearest whole number.)

𝛼 = 15°, b = 12

To solve these problems, we can use the trigonometric ratios for right triangles: sine, cosine, and tangent.

1) To approximate the remaining parts of triangle ABC with γ = 90°, β = 51°31', and b = 250.0:

a) To find α, we can use the sine ratio: sin(α) = opposite/hypotenuse. In this case, the opposite side is b and the hypotenuse is c. Since we don't have the value of c, we won't be able to directly find α.

b) To find a, we can use the cosine ratio: cos(β) = adjacent/hypotenuse. In this case, the adjacent side is a and the hypotenuse is c. Rearranging the equation, we get: a = cos(β) * c. Since we don't have the value of c, we won't be able to directly find a.

c) We can use the Pythagorean theorem to find c, since it is the hypotenuse: c^2 = a^2 + b^2. Rearranging the equation, we get: c = sqrt(a^2 + b^2). Now, we can substitute the given values and approximate the solution.

2) To approximate the remaining parts of triangle ABC with γ = 90°, α = 27°50', and a = 530:

a) To find β, we can use the sine ratio: sin(β) = opposite/hypotenuse. In this case, the opposite side is b and the hypotenuse is c. Since we don't have the value of c, we won't be able to directly find β.

b) To find b, we can use the cosine ratio: cos(α) = adjacent/hypotenuse. In this case, the adjacent side is b and the hypotenuse is c. Rearranging the equation, we get: b = cos(α) * c. Since we don't have the value of c, we won't be able to directly find b.

c) We can use the Pythagorean theorem to find c, since it is the hypotenuse: c^2 = a^2 + b^2. Rearranging the equation, we get: c = sqrt(a^2 + b^2). Now, we can substitute the given values and approximate the solution.

3) To express the third part in terms of the first two parts, given β, c, and b:

We can use the sine ratio: sin(β) = opposite/hypotenuse. In this case, the opposite side is b and the hypotenuse is c. Rearranging the equation, we get: b = sin(β) * c.

4) To express the third part in terms of the first two parts, given α, a, and b:

We can use the cosine ratio: cos(α) = adjacent/hypotenuse. In this case, the adjacent side is b and the hypotenuse is a. Rearranging the equation, we get: b = cos(α) * a.