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Law of Sines

Law of Sines. Given the triangle below …. … the law of sines is given by …. Law of Sines. Note that in each ratio, the sine of the angle is written over the length of the side opposite that angle.

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Law of Sines

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  1. Law of Sines • Given the triangle below … • … the law of sines is given by …

  2. Law of Sines • Note that in each ratio, thesine of the angle is written over the length of the side opposite that angle. • Note also that the triangle is not a right triangle, so the pythagorean theorem cannot be used.

  3. Example 1: • Solve the triangle with the given measures: • Since the three angles of a triangle add up to 180 degrees …

  4. Now use the law of sines. Since side b is given, one ratio will include side band angleβ(the angle opposite side b). • The other ratio is our choice since we know the value of both angles.

  5. Note: make sure the calculator is set to degree mode.

  6. Now find the last side c. Use side b for the other ratio since it is given. Using the rounded value of a would lead to further rounding error.

  7. All missing measures have been found and the triangle is solved.

  8. Example 2: • Solve the triangle with the given measures: • Since side b and angleβare both given, use them for one ratio. • Since side a is given, use it for the other ratio.

  9. Note: make sure the calculator is set to degree mode.

  10. We know two angles now, so find the third angle.

  11. Find the remaining side.

  12. All missing measures have been found and the triangle is solved.

  13. Before we move on, consider one of the calculations that was made in this problem. • The calculator gave us a value of …

  14. Consider a unit circle with the given information. • There is another possible value for the angle. Using a reference angle of 71 degrees … • … we find another angle that will also solve the equation. • Note that 109 degrees is possible for an angle in a triangle.

  15. Recall the sides and angles that have been found up to this point • Given: • Determined:

  16. Now consider a second possible triangle. • Use the second value found for α and re-solve the triangle.

  17. The given information led to two different triangles.

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