Investigation: Deriving the Sine Law

Glossary Checklist FAQ

Deriving the Sine Law

1. Using the triangle on the left, write the trigonometric ratio you would use to find side \(h\).
   Do not evaluate the ratio at this point!
   
   
2. Solve for \(h\), using the triangle on the left.
   
   
3. Using the triangle on the right, write the trigonometric ratio you would use to find side \(h\).
   Do not evaluate the ratio at this point!
   
   
4. Solve for \(h\), using the triangle on the right.
   
   
5. Compare the expressions you found for side \(h\). What do you notice?
   
   
   Use the triangle and information below to answer the next question.
   
   
   If you follow the same procedure for this triangle, then the left side will give you:
   
   \(\begin{align}
    \sin \theta &= \frac{{{\mathop{\rm opp}\nolimits} }}{{{\mathop{\rm hyp}\nolimits} }} \\
    \sin 67^\circ &= \frac{h}{{46}} \\
    46 \sin 67^\circ &= h \\
    \end{align}\)
   
   Similarly, the right-hand side will give you:
   
   \(\begin{align}
    \sin \theta &= \frac{{{\mathop{\rm opp}\nolimits} }}{{{\mathop{\rm hyp}\nolimits} }} \\
    \sin 35^\circ &= \frac{h}{a} \\
    a \sin 35^\circ &= h \\
    \end{align}\)
   
   Because both equations equal h, you can make them equal each other.
   
   \(46 \sin 67^\circ = a \sin 35^\circ \)
   
   
6. Using this new equation, solve for \(a\).
   
   Notice that, instead of solving first for \(h\), and then using the value for \(h\) to determine side \(a\), you go directly to finding side \(a\). Handy!
   
   
   Use the diagram below to answer the next question.
   
   
   
   
7. Looking at the equation you used in question 6, what value(s) have changed? Change them and determine the length of side \(a\).
   
   
   Use the diagram below to answer the next question.
   
   
   
   
8. Looking at the equation you used in question 6, substitute the numbers in the equation with the corresponding variables found in the above diagram.
   
   
   
   
   ...continued on the next page