Question

Find the missing side length and angles of △ABC given that m∠C=129∘, a=7, and b=10.

Answer 1

To solve for the missing side length and angles of triangle ABC, we can use the Law of Cosines and the Law of Sines.

First, we can use the Law of Cosines to solve for side c:

c^2 = a^2 + b^2 - 2ab cos(C)

c^2 = 7^2 + 10^2 - 2(7)(10) cos(129°)

c^2 ≈ 39.37

c ≈ 6.27

So the length of side c is approximately 6.27.

Next, we can use the Law of Sines to solve for the remaining angles:

sin(A)/a = sin(C)/c

sin(A)/7 = sin(129°)/6.27

sin(A) ≈ 0.564

A ≈ 34.5°

To find angle B, we can use the fact that the sum of the angles in a triangle is 180°:

B = 180° - A - C

B ≈ 16.5°

Therefore, the missing side length is approximately 6.27 and the missing angles are approximately A = 34.5° and B = 16.5°.

Answer 2

Explanation:

To find the missing side length and angles of triangle ABC, given that angle C is 129°, side a is 7 units, and side b is 10 units, we can use the Law of Cosines and the Law of Sines.

Finding side c using the Law of Cosines:

According to the Law of Cosines, in a triangle ABC, c^2 = a^2 + b^2 - 2abcos(C).

Substituting the given values: c^2 = 7^2 + 10^2 - 2(7)(10)cos(129°).

Calculating: c^2 ≈ 49 + 100 + 140cos(129°).

Simplifying: c^2 ≈ 149 + 140(-0.64278760968).

Calculating: c^2 ≈ 149 - 89.99852275552.

Therefore, c^2 ≈ 59.00147724448.

Taking the square root: c ≈ √59.00147724448.

Hence, c ≈ 7.68 units (rounded to two decimal places).

Finding angle A using the Law of Sines:

According to the Law of Sines, sin(A)/a = sin(C)/c.

Substituting the given values: sin(A)/7 = sin(129°)/7.68.

Cross-multiplying: 7sin(A) = 7.68sin(129°).

Calculating: sin(A) ≈ (7.68sin(129°))/7.

Taking the inverse sine: A ≈ sin^(-1)((7.68sin(129°))/7).

Hence, A ≈ 49.3° (rounded to one decimal place).

Finding angle B:

Since the sum of the angles in a triangle is 180°, angle B = 180° - angle A - angle C.

Substituting the given and calculated values: B = 180° - 49.3° - 129°.

Calculating: B ≈ 1.7° (rounded to one decimal place).

Therefore, in triangle ABC:

Side a = 7 units

Side b = 10 units

Side c ≈ 7.68 units (rounded to two decimal places)

Angle A ≈ 49.3° (rounded to one decimal place)

Angle B ≈ 1.7° (rounded to one decimal place)

Angle C = 129°