Law of Sines & Cosines: Formula & Examples

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Solving trigonometry’s problem that use right angle can be done by using right triangle and Pythagorean theorem. But, in other problems it does not use right triangle.

It uses any triangles. The solution is using law of sines and cosines. It will be more complex than using right triangle but it does not mean difficult.

Law of Sines

Law of sines use to solve triangle’s problem if known a pair of angles – sides and another side or angle and it will be found the pair.

In another words, law of sines can be applied if a triangle has one of the pair (in a row):

  • Angle – angle – side
  • Angle – side – angle
  • Side – side – angle

In mathematics formula, if there is a triangle, the law of sines can be written

Law of sinus

sinα/a = sinβ/b = sinγ/c

or

a/sin α = b/sin β = c/sin γ

(choose one of equation form depend on the problem. Just use two of the equation based on known in the problem)

Law of Cosines

Law of cosines is also used if there is no right triangle and the problem is finding the angle or side. It is different with law of sines. Law of cosines can be used if known:

  • Side – angle – side
  • Side – side – side

In mathematics, if there is any triangle, then

Law of cosinus

a2 = b2 + c2 – 2.a.b.cos α

or

b2 = a2 + c2 – 2.a.c.cos β

or

c2 = a2 + b2 – 2.a.b.cos γ

(choose one of equation depend on the problem)

note:

remember that the sum of angles in triangle is 180°. So, if there is known two angles then the third have been known too.

Examples

  1. There is triangle like the picture. Find the value of x.
Law of sinus example
Answer

Known angle – side – side or side – side – angle then using law of sines to solve it

      sin α / a = sin β / b = sin γ / c

sin 30° / 10 = sin x / 8

      ½ / 10 = sin x / 8

              4 = 10 . sin x

         Sinx = 0,4

               x = arc-1sin(0,4)

2. There is triangle. Find the value of x.

Law of cosinus example
Answer

First step: determine the value of α

180° – (105° + 60°) = 15°

Next step: use law of sines

        Sin15°/4 = sin105°/x

{¼ (√6-√2)}/4 = {¼ (√6+√2)}/x

                {x(√6-√2)}/4 = (√6+√2)

x(√6-√2) = 4(√6+√2)

             x = {4(√6+√2)} / (√6-√2)

             x = (√6-+√2)2

Note:

sin15° = sin(45°-30°) = sin45.cos30 – cos45°.sin30° 

= (½ √2)( ½ √3) – (½ √2)( ½)

= ¼√6 – ¼√2

= ¼ (√6-√2)

Sin105° = sin(180° – 75°) = sin75°

Sin75° = sin(45°+30°) = sin45.cos30 + cos45°.sin30° 

= (½ √2)( ½ √3) + (½ √2)( ½)

= ¼√6 + ¼√2

= ¼ (√6+√2)

3. There is triangle that has 6cm, 8cm, and 12cm as the sides. Determine the value of cosines of smallest angle in the triangle.

Answer

Smallest angle (let it as α) is in front of shortest side. It is in front of 6cm side.

Using law of cosines, it will be gotten:

              62 = 82 + 122 – 2.8.12.cosα

             36 = 64 + 144 – 192.Cosα

192.Cosα = 208 – 36

       Cosα = 172/192

       Cosα = 0,89