Stonehenge is believed to have been constructed 5000 years ago in three phases. There is a lot of mystery around Stonehenge involving questions about why and how it was built. For mathematicians, it is very interesting to study due to the fact that its builders appeared to have used right triangles at least 2,000 years before Pythagoras came up with his famous theorem. One interesting thing to study is the way the circle of stones is aligned with the angle of the sun on summer solstice. On the summer solstice, the sun is directly in line with the heel stone (outside of the circle), the avenue into the circle, and the horseshoe shape of stones on the inside of the circle. |
1) Given the diagram of Stonehenge, write out the six trig ratios for the angle of the summer solstice. Round the length of the hypotenuse to the nearest whole number to include in your ratios.
- \(\sin\theta=?\)
- \(\cos\theta=?\)
- \(\tan\theta=?\)
- \(\csc\theta=?\)
- \(\sec\theta=?\)
- \(\cot\theta=?\)
2) The reference angles for the sunrise and sunsets of both the summer and winter solstices are all the same. That means that the vertical dimension of \(35\) ft and the horizontal dimension of \(41\) ft will both apply to all four positions on the circle. Based on the given trig ratios, decide whether the information is referring the winter solstice sunrise, summer solstice sunrise, winter solstice sunset, or summer solstice sunset.
A) \(\sin\theta=-\frac{35}{54}\ \&\ \cos\theta=\frac{41}{54}\) Worked Out Example
B) \(\tan\theta=-\frac{35}{41}\ \&\ \sin\theta=\frac{35}{54}\)
C) \(\cos\theta=-\frac{41}{54}\ \&\ \tan\theta=\frac{35}{41}\)
D) \(\csc\theta=\frac{54}{35}\;\;\;\ \&\ \cot\theta=\frac{41}{35}\)
Solutions