The time has come to issue a warning: this entire series of caches will be archived between May 1st and October 31st, 2022.
Congratulations to Opalline and Yugsirap for their FTF achievement
- even if they really wanted the glory of being STF!
Caches in the Series
The series consists of: Archimedes (GC3TT8C), Bernoulli (GC3V9MW), Cayley (GC3W898), Descartes (GC3X0W7), Euclid (GC3XGN0), Fermat (GC413V5), Gauss (GC4219Z), Heron (GC4VGBX), Ingham (GC4WC72), Jarnik (GC4Z9PK), Kepler (GC4ZMRG), Lovelace (GC50BN6), Merryman (GC51D34) and Newton (GC5AJ04).
Introduction to the Cache
For this cache, you will need some sort of scientific calculator whether it be on the computer, on your “smart ‘phone” or one of those ten-dollar portable items we used at school. The entire cache is intended to refresh your memory in how to handle the calculator with respect to the use of the ratios sine, cosine and tangent.
Fermat the Mathematician
Along with René Descartes, seventeenth century Pierre de Fermat has the reputation for being one of the most influential mathematicians of his era. Fermat was actually a lawyer and a civil servant so for him, mathematics was only a hobby. However, he practically founded number theory and had much to do with the discoveries of analytic geometry and calculus. He was also an excellent geometrician and assisted Pascal in discovering probability theory. His name is given to the international mathematics competition provided annually for grade eleven students by Canada’s University of Waterloo.
Fermat the Cache
At some time during the working out of the co-ordinates for this cache, you may find it useful to look back at Descartes, the fourth cache in this series, with respect to the special ratios - sine, cosine and tangent. So useful are these ratios that their values have been calculated to many decimal places and, in the past, those wishing this information would consult books of tables - and that is still possible. More common now, is the use of a computer or any one of a variety of cheap “scientific” calculators - so cheap that many students think little of leaving them in a locker at the end of a school year! It will be the use of calculators - on-line or in-hand - to determine these ratios that will be the means to finding the co-ordinates of this cache.
Use of a Calculator to Find Sine, Cosine and Tangent
The Calculator Itself
So-called “scientific” calculators have buttons on them marked “sin,” “cos” and “tan.” In order to use the calculator as we are going to do it, one must ensure that the calculator is adjusted for “degrees” (deg) rather than “radians” (rad) or “gradients” (grad.). It is a simple adjustment and differs slightly from calculator to calculator. That having been done, one finds the required ratio (sine, cosine or tangent) of a given angle by simply punching in the size of the angle and pushing the appropriate button. The required ratio will appear on the screen in decimal form.
We will be looking for the ratios to the nearest thousandth only. Examples are: 0.978192 will be rounded off to 0.978 whereas 0.978754 will become 0.979; 2.832499 we will treat as 2.832 but 2.832501 will become 2.833.
Abbreviations
Note that:
➢ ≐(an equals sign with a dot over it) means “approximately equal” and will be used to indicate either a ratio or an angle that has been rounded off.
➢ m° (the letter m with a degrees symbol) means “the degree measure of” and will be followed by a letter representing an angle such as: m°∡M, meaning “the degree measure of angle M.”
Test Yourself
sin 21° ≐ _______; cos 67° ≐ _______; tan 82° ≐ _______;
cos 17.31° ≐ _______; sin 32.59° ≐ ________. (0.358, 0.391, 7.115, 0.955, 0.539)
Finding the Angle Given the Sine, Cosine or Tangent
The Process
The calculator has a button - often at the top left of the “keyboard” - labelled “2ndF” which stands for "second function" which implies that a button can do two things. Using the "2ndF" button will cause the calculator do what is marked not on the button, but above it on the body of the calculator. In this case the “sin” button will have above it "sin-1" meaning “input the sine and the display will give the angle.” So, to determine an angle given its sine, you punch in the given sine and then press “2ndF” followed by “sin” and the angle will appear.
Try These! (Round the angles off to the nearest tenth - i.e. one decimal place.)
sin ∡M = 0.5894, so m°∡M ≐ ____; cos ∡N = 0.6255, so m°∡N ≐ ____;
tan ∡P = 0.8888, so m°∡P ≐ ____; tan ∡Q = 3.2613, so m°∡Q ≐ ____.
(Answers: 36.1, 51.3, 41.6, 73.0)
And Now . . . the Cache!
The “Puzzle”
1. Find the following, rounding off to three decimal places:
(a) sin 47.2° ≐ _ .A _ _ ; (b) cos 19.0° ≐ _ .B _ C;
(c) cos 73.7° ≐ _ . _ _ D; (d) tan 80.3° ≐ _ . E _ F
2. Find the following, rounding angles off to one decimal place:
(a) sin ∡P = 0.234 so m°∡P ≐ _ G.H; (b) sin ∡Q = 0.890 so m°∡Q ≐ _ J. _ ;
(c) cos ∡R = 0.678 so m°∡R ≐ K _. L; (d) tan ∡S = 1.111 so m°∡S ≐ _ M. _.
In addition, N is the ones unit of the west co-ordinate of the location of the cache in Euclid.
A = ___; B = ___; C = ___; D = ___; E = ___; F = ___; G = ___;
H = ___; J = ___; K = ___; L = ___; M = ___; N = ___.
The Co-ordinates
The co-ordinates are: N 44°ab.cde’ and W 078° fg.hij’ where these lower case letters are unrelated to any letters used previously.
a = B - M = __;
b = C + K - J = __;
c = 2(L + N) - E = __;
d = A - F = __;
e = D + G - F = __;
f = M - G = __;
g = B - A = __;
h = N + J = __;
i = D + H = __;
j = 2L + C - H = __.
So the cache is to be found at: N 44° _ _ . _ _ _’ and W 078° _ _ . _ _ _’.
Other Comments
- Please bring your own writing instrument.
- The cache is a well-washed peanut butter container.
- Also, the difficulty rating is not due to the hide.
Check Your Answer
You can check your answers for this puzzle on GeoChecker.com.