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TrianglesGeometryHigh schoolabout 2:01

Two sides of a triangle are 10 and 12 with a 75° angle between them. Find the third side: animated lesson

A narrated, step-by-step triangles video lesson with captions. Final answer: c ≈ 13.486416; A ≈ 45.743376°, B ≈ 59.256624°.

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.8 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    Two sides of a triangle are 10 and 12 with a 75° angle between them. Find the third side.
  2. 2. Steps, part 1
    1. c2=102+122−2(10)(12)cos⁡75∘≈181.88343c^2 = 10^2 + 12^2 - 2(10)(12)\cos 75^\circ \approx 181.88343
      Substitute into c² = a² + b² − 2ab·cos C
    2. c≈13.486416c \approx 13.486416
      Take the square root
    3. cos⁡A=b2+c2−a22bc≈0.69787327⇒A≈45.743376∘\cos A = \frac{b^2 + c^2 - a^2}{2bc} \approx 0.69787327 \Rightarrow A \approx 45.743376^\circ
      Find angle A with the law of cosines rearranged
    4. B=180∘−A−C≈59.256624∘B = 180^\circ - A - C \approx 59.256624^\circ
      Angles of a triangle add to 180°
  3. 3. Steps, part 2
    1. Area=12absin⁡C≈57.95555\text{Area} = \tfrac12ab\sin C \approx 57.95555
      Area from two sides and the included angle
  4. 4. The answer
    c ≈ 13.486416; A ≈ 45.743376°, B ≈ 59.256624°\text{c ≈ 13.486416; A ≈ 45.743376°, B ≈ 59.256624°}

    c ≈ 13.486416; A ≈ 45.743376°, B ≈ 59.256624°

  5. 5. Check
    • A + B + C = 180°.
    • Heron's formula gives the same area: 57.95555.
    • Law of sines ratios: a/sin A ≈ 13.962166, b/sin B ≈ 13.962166, c/sin C ≈ 13.962166.

Transcript

The problem. Here is the problem: Two sides of a triangle are 10 and 12 with a 75° angle between them. Find the third side. SAS: find the third side and the remaining angles.

Steps, part 1. Substitute into c squared equals a squared plus b squared minus 2ab times cos C. Take the square root. Find angle A with the law of cosines rearranged. Angles of a triangle add to 180°.

Steps, part 2. Area from two sides and the included angle.

The answer. So the answer is c approximately 13.486416; A approximately 45.743376°, B approximately 59.256624°. The triangle has sides a equals 10, b equals 12, c approximately 13.486416 and angles A approximately 45.743376°, B approximately 59.256624°, C approximately 75°. It is an acute scalene triangle with area approximately 57.95555 and perimeter approximately 35.486416.

Check. A plus B plus C equals 180°. Heron prime s formula gives the same area: 57.95555. Law of sines ratios: a/sin A approximately 13.962166, b/sin B approximately 13.962166, c/sin C approximately 13.962166.

About triangles

Solve triangles with the law of sines and the law of cosines, and find their area.

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