In △ABC, m∠A=α, m∠B=β, m∠C=γ. AB=c, AC=b, BC=a. Find the remaining parts of each triangle if the following parts are given.
a) a=4.60, b=3.10, α=18º. Find: c≈__, β≈__, γ≈__
b) a=5.24, b=4.44, β=48º. Find: c≈ __, α≈ ___, γ≈ __
a) sinA/a = sinB/b
Now you have α and β, so γ is easy
the use the law of sines again to get c.
b) same method. Use law of sines
Ty
a) Given a=4.60, b=3.10, and α=18º, we need to find c, β, and γ.
To find c, we can use the Law of Cosines:
c² = a² + b² - 2ab cos(α)
Substituting the given values:
c² = (4.60)² + (3.10)² - 2(4.60)(3.10) cos(18º)
Solving for c:
c ≈ √[(4.60)² + (3.10)² - 2(4.60)(3.10) cos(18º)]
Now, to find β, we can use the Law of Sines:
sin(β) / b = sin(α) / a
Substituting the given values:
sin(β) / 3.10 = sin(18º) / 4.60
Solving for sin(β), and then β:
sin(β) ≈ (3.10 * sin(18º)) / 4.60
β ≈ sin⁻¹((3.10 * sin(18º)) / 4.60)
Lastly, to find γ, we can use the fact that the angles in a triangle add up to 180º:
γ ≈ 180º - α - β
b) Given a=5.24, b=4.44, and β=48º, we need to find c, α, and γ.
To find c, we can use the Law of Cosines:
c² = a² + b² - 2ab cos(β)
Substituting the given values:
c² = (5.24)² + (4.44)² - 2(5.24)(4.44) cos(48º)
Solving for c:
c ≈ √[(5.24)² + (4.44)² - 2(5.24)(4.44) cos(48º)]
Now, to find α, we can use the Law of Sines:
sin(α) / a = sin(β) / c
Substituting the given values:
sin(α) / 5.24 = sin(48º) / c
Solving for sin(α), and then α:
sin(α) ≈ (5.24 * sin(48º)) / c
α ≈ sin⁻¹((5.24 * sin(48º)) / c)
Lastly, to find γ, we can use the fact that the angles in a triangle add up to 180º:
γ ≈ 180º - α - β
To find the remaining parts of each triangle, it is necessary to use the properties and formulas of triangles, specifically the Law of Cosines and the Law of Sines.
a) Given a=4.60, b=3.10, and α=18º, we need to find c, β, and γ.
1. To find c, we can use the Law of Cosines, which states that c² = a² + b² - 2ab cos(γ). Rearranging the formula, we have cos(γ) = (a² + b² - c²) / (2ab).
Using the given values and α, we can substitute them into the equation:
cos(γ) = (4.60² + 3.10² - c²) / (2 * 4.60 * 3.10)
2. Now we can find γ using the inverse cosine function (cos⁻¹).
γ ≈ cos⁻¹ [(4.60² + 3.10² - c²) / (2 * 4.60 * 3.10)]
3. To find β, we can use the Law of Sines, which states that sin(β) / b = sin(γ) / c.
Substituting in the known values, we get sin(β) / 3.10 = sin(γ) / c.
We can now solve for β using the equation:
β ≈ sin⁻¹ [(3.10 * sin(γ)) / c]
b) Given a=5.24, b=4.44, and β=48º, we need to find c, α, and γ.
1. To find c, we can again use the Law of Cosines: c² = a² + b² - 2ab cos(γ).
Substituting the known values, we have c² = 5.24² + 4.44² - 2 * 5.24 * 4.44 * cos(γ).
2. To find γ, we can rearrange the equation and use the inverse cosine function:
γ ≈ cos⁻¹ [(5.24² + 4.44² - c²) / (2 * 5.24 * 4.44)]
3. Lastly, to find α, we can use the Law of Sines: sin(α) / a = sin(γ) / c.
Substituting the known values, we get sin(α) / 5.24 = sin(γ) / c.
We can now solve for α:
α ≈ sin⁻¹ [(5.24 * sin(γ)) / c]
To find approximate values for c, β, γ, α, we can plug the known values into the respective equations and calculate the results.