Using trigonometric ratios find the remaining side lengths and angle measurements to 2 decimal places each:
(a) α = 35°, β = 90°, a = 30
(b) α = 90°, b = 7, c = 6
(c) a = 33, b = 56, c = 65
(a) γ is easy, since the angles sum to 180°
Then use law of sines to find b,c
(b) this is a right triangle. a^2 = b^2+c^2
then just use the trig functions to get the other two angles.
(c) use the law of cosines to get any or all of the angles. Or, having one, you can use the law of sines to get the others.
To find the remaining side lengths and angle measurements, we can use the trigonometric ratios: sine, cosine, and tangent. Let's go through each problem step by step.
(a) In triangle ABC, we are given α = 35°, β = 90°, and a = 30.
To find side b, we can use the sine ratio:
sin(α) = b / a
sin(35°) = b / 30
b = 30 * sin(35°) ≈ 17.23
To find side c, we can use the Pythagorean theorem:
c² = a² + b²
c² = 30² + 17.23²
c ≈ √(900 + 297.2729)
c ≈ 34.21
To find angle γ, we can use the fact that the angles in a triangle add up to 180°:
γ = 180° - α - β
γ = 180° - 35° - 90°
γ ≈ 55°
So, for triangle ABC, b ≈ 17.23, c ≈ 34.21, and γ ≈ 55°.
(b) In triangle ABC, we are given α = 90°, b = 7, and c = 6.
To find side a, we can use the Pythagorean theorem:
a² = c² - b²
a² = 6² - 7²
a ≈ √(36 - 49)
a ≈ √(-13) (Note: The square root of a negative number is not defined in the real number system. Therefore, triangle ABC with these given side lengths is not possible.)
(c) In triangle ABC, we are given a = 33, b = 56, and c = 65.
To find angle α, we can use the cosine ratio:
cos(α) = (b² + c² - a²) / (2 * b * c)
cos(α) = (56² + 65² - 33²) / (2 * 56 * 65)
cos(α) = (3136 + 4225 - 1089) / (2 * 56 * 65)
cos(α) ≈ 7288 / (2 * 56 * 65)
cos(α) ≈ 0.5579
α ≈ arccos(0.5579)
α ≈ 56.99°
To find angle β, we can use the sine ratio:
sin(β) = a / c
sin(β) = 33 / 65
β ≈ arcsin(0.5077)
β ≈ 30.71°
To find angle γ, we can use the fact that the angles in a triangle add up to 180°:
γ = 180° - α - β
γ = 180° - 56.99° - 30.71°
γ ≈ 92.3°
So, for triangle ABC, α ≈ 56.99°, β ≈ 30.71°, and γ ≈ 92.3°.