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Lesson · Grade 11 · Pre Calculus

The cosine law: two sides and the angle between them

  • Grade 11
  • Pre Calculus
  • 2 min 26 s
  • British Columbia
  • BC11-MA-80

One labelled triangle (b = 8 cm, c = 11 cm, included angle A = 37°) carries the whole video: it introduces a² = b² + c² − 2bc cos A, shows that cos 90° = 0 turns it back into the Pythagorean theorem, and says when to use it. It then works the example step by step to a² ≈ 185 − 140.56 ≈ 44.44, so a ≈ 6.67 cm, shows the common mistake of subtracting before multiplying by cos A (which gives 7.19), checks the answer against the 90° case (√185 ≈ 13.6 cm), and ends on a one-line recap.

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In one sentence

The question this lesson answers

This triangle has sides of 8 and 11 centimetres, with a 37 degree angle between them. How long is the third side?

Each side takes the small letter of the opposite angle, so side a is opposite angle A.

Transcript

What the lesson says

This triangle has sides of 8 and 11 centimetres, with a 37 degree angle between them. How long is the third side?

Each side takes the small letter of the opposite angle, so side a is opposite angle A.

Angle A lies between sides b and c, so it is called the included angle.

There is no right angle, so we use the cosine law: a squared equals b squared plus c squared, minus 2 b c cosine A.

The last part is a correction. At 90 degrees, cosine A is 0, so it drops out, leaving the Pythagorean theorem: a squared equals b squared plus c squared.

Use it when you know two sides and the included angle, or when you know all three sides.

For our triangle, put in b equals 8, c equals 11, and A equals 37 degrees.

8 squared is 64, 11 squared is 121, and 2 times 8 times 11 is 176.

Cosine 37 degrees is about 0.79864, so 176 times cosine 37 degrees is about 140.56.

64 plus 121 is 185, and 185 minus 140.56 is about 44.44.

Take the square root: side a is about 6.67 centimetres.

A common mistake is to work out 185 minus 176 first, then multiply by cosine 37 degrees. That gives 7.19, which is wrong.

Always multiply by cosine A first, then subtract.

Check: at 90 degrees, side a would be the square root of 185, about 13.6 centimetres. 37 degrees is less, so side a is shorter.

So with two sides and the included angle, the cosine law gives the third side.

Curriculum

Where this fits

British Columbia, Grade 11

Pre Calculus 11, Trigonometry: trigonometry: non-right triangles and angles in standard positionBC11-MA-80

Previous: The Sine Law: How Far Away Is the Fire?Next: The Ambiguous Case: Two Triangles from the Same Facts

The outcomes and strands follow the British Columbia curriculum.

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