Compare Your Answers
  1. What similarities do you notice between the sine and cosine values for \(30^\circ\), \(45^\circ\), and \(60^\circ\)?

    sin \(30^\circ\) = cos \(60^\circ\) and sin \(60^\circ\) = cos \(30^\circ\)

    sin \(45^\circ\) = cos \(45^\circ\)

  2. What similarities do you notice between the tangent values for \(30^\circ\) and \(60^\circ\)?

    The numerator and denominator values have switched places.

  3. BONUS: What do you notice when you divide sin \(\theta\) by cos \(\theta\)? Compare the result to tan \(\theta\) for \(30^\circ\), \(45^\circ\), and \(60^\circ\).

    \(\begin{align}
     \sin 30^ \circ  \div \cos 30^ \circ  &= \frac{1}{2} \div \frac{{\sqrt 3 }}{2} \\
      &= \frac{1}{2} \cdot \frac{2}{{\sqrt 3 }} \\
      &= \frac{1}{{\sqrt 3 }} \\
      &= \frac{{\sqrt 3 }}{3} \\
      &= \tan 30^ \circ   \\
     \tan 30^ \circ  &= \sin 30^ \circ \div \cos 30^ \circ   \\
     \end{align}\)
    \(\begin{align}
     \sin 45^ \circ  \div \cos 45^ \circ &= \frac{{\sqrt 2 }}{2} \div \frac{{\sqrt 2 }}{2} \\
      &= \frac{{\sqrt 2 }}{2} \cdot \frac{2}{{\sqrt 2 }} \\
      &= \frac{{2\sqrt 2 }}{{2\sqrt 2 }} \\
      &= 1 \\
      &= \tan 45^ \circ   \\
     \tan 45^ \circ &= \sin 45^ \circ \div \cos 45^ \circ   \\
     \end{align}\)
    \[\begin{align}
     \sin 60^ \circ  \div \cos 60^ \circ &= \frac{{\sqrt 3 }}{2} \div \frac{1}{2} \\
      &= \frac{{\sqrt 3 }}{2} \cdot \frac{2}{1} \\
      &= \frac{{2\sqrt 3 }}{2} \\
      &= \sqrt 3  \\
      &= \tan 60^ \circ   \\
     \tan 60^ \circ &= \sin 60^ \circ \div \cos 60^ \circ   \\
     \end{align}\]

    Therefore, \(\tan \theta = \sin \theta \div \cos \theta \) or \({\rm{tan}}\theta = \frac{{\sin \theta }}{{\cos \theta }}\).