- Difficulty:

- Terrain:

Size: (regular)

The caches in this series have to do with triangles. The “learning” is largely cumulative so, in general, one is advised to work in alphabetical order of the caches which are named after famous mathematicians. However, reviewing the instructions for Descartes (GC3X0W7) and Fermat (GC413V5), and using the information included below should be sufficient for this 2.7 km round trip **"Field Puzzle Multi."**

**Congratulations to Opal & Yugsilrap on earning yet another FTF!!**

It would have been fun to be a bird following along!

**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).

**Merryman - the Person**

Jerry D. Merryman started work with *Texas Instruments* in 1963 and stayed with them for thirty years. He, along with James H. Van Tassel and the late Jack S. Kilby were responsible for the invention of the handheld electronic calculator and it is for this reason that this cache carries his name.

Other areas in which he worked were: designing and creating fabricating semiconductor manufacturing equipment, contributing to the thermal printing devices that the firm used for years and working in the area of industrial automation. Obviously a man with a creative imagination, he holds approximately sixty patents and, at this writing, is still a consultant to *Texas Instruments.*

**Merryman - the Cache**

**The Essence of this Cache: The Use of a Scientific Calculator in the Field**

We are familiar from previous caches in this series of the use of a scientific calculator in determining the sine, cosine and tangent of an angle and, given one of those ratios, finding the angle related to it (see “Descartes” and “Fermat” as mentioned above). However, we have not worked with the so-called “reciprocal ratios” nor the use of the calculator in working with them.

**Reciprocal Ratios**

The **reciprocal** of a fraction is simply the **fraction turned upside down**.

The **sine (sin)** of an angle is a fraction defined as the **length of the opposite side over the length of the hypotenuse**; the **cosecant (csc)** is defined as the **hypotenuse over the opposite** - i.e. the reciprocal of the sine.

The **cosine (cos)** is a fraction defined as the **adjacent over the hypotenuse**; the **secant (sec)** is defined as the **hypotenuse over the adjacent** - i.e. the reciprocal of the cosine.

The **tangent (tan)** is a fraction defined as the **opposite over the adjacent**; the **cotangent (cot)** is defined as the **adjacent over the opposite** - i.e. the reciprocal of the tangent.

So, if you are asked for a reciprocal ratio (csc, sec or cot) of a specific angle, find the primary ratio (sin, cos, tan) as we have done before and invert it. And that’s all there is to it!!

**Examples**

(a) sin A is 2/5 or 0.4; csc A is 5/2 or 2.5.

(b) cos K is 1/2 or 0.5; sec K is 2/1 or 2.0.

(c) tan W is 28/55 or approximately 0.509; cot W is 55/28 or approximately 1.964.

**The Use of a Calculator to Change from Primary to Reciprocal Ratios and the Reverse**

If you are given one of the six ratios as a fraction (e.g. 3/5), you do not need a calculator to find the reciprocal ratio - you simply turn the fraction over (e.g. 5/3). Of course, if you want that ratio in decimal form, you need to divide the numerator (top) of the ratio by the denominator (bottom) and you will get 0.6 or 1.66 . . . respectively in these cases.

More often, however, you will be given the ratio as a decimal. Calculators have buttons for the primary ratios - sine, cosine and tangent - marked sin, cos and tan respectively. They do not have buttons for the reciprocal ratios. So what you have to do if you are asked for the cosecant (or secant or cotangent) of an angle is to find the sine (or cosine or tangent) and turn it over - and there is a button on the calculator to do that. It is indicated by 1/x and it may be right on the button in which case you have only to push the button. It may, however, be marked on the body of the calculator and so you have to push the “shift” sometimes called “2nd function” button first. Let’s try an example or two.

**Examples**

(a) Suppose that you are given that the sine of a certain angle, B, is 0.653 and you are asked for the cosecant of B. Once you have 0.653 in the calculator, push the 1/x button - using the “shift” key or not depending upon where the 1/x is printed. You will find that csc B ≐ 1.531.

(b) cos M = 0.93248; so sec M ≐ ____ (1.0724).

(c) tan D = 3.876; so cot D ≐ _____ (0.2580).

(d) csc P = 8.345; so sin P ≐ _____ (0.1198).

(e) sec F = 43.012; so cos F ≐____ (0.0232).

(f) cot N = 0.3456; so tan N ≐ ____ (2.8935).

**To Find an Angle, Given a Reciprocal Ratio**

We have done this with primary ratios before, but you may wish to review the process in “Fermat.”

You need first to ensure that you have the “mode” of your calculator on degrees; the display should show “deg” or something of that sort.

If given the csc (or sec or cot) of an angle, you need first to use the 1/x button to change to the sin (or cos or tan). Then you do exactly what we did in “Fermat.”

**Examples**

(a) csc A = 2.503 so A ≐ ____ (23.5° )

(b) sec B = 5.792 so B ≐ ____ (80.06°)

(c) cot C = 0.875 so C ≐ _____ (48.8° ).

**The Cache Hunt!**

Go to the stated co-ordinates. **Be sure to take some paper, a pen and a scientific calculator with you** - be it in your GPSr, cell phone or otherwise. This is the first stage of a **four-stage field puzzle multi**. The first three stages are pill bottles; the final is a "regular" and might be tricky to find during a hard winter. The solution of the field puzzle at stage one will take you to stage two and so on. Don’t round off - let your calculator do the work!! Good luck!!

**Optional Supplement: Finding the Given Stage One**

**Introduction**

This section is to give you some idea of what you will find in the field to help you move on through the four stages. You are given questions (below) similar to those you will find at stages one, two and three but **you already know the answer - the given co-ordinates for Stage One as given as the "stated co-ordinates" for this cache**. You are advised to work it through to ensure that you get the right answer - i.e. that you have the right idea of what to do "in the field." Then, if you’re having a problem, you can contact someone - the CO, if you like - to find out where you are going wrong. It would be too bad to find yourself in the field and unable to get yourself to Stage Two!

**The Sort of Thing You Will Find at Each of Stages One, Two and Three**

You are looking for N 44° 19.ABC’ and W 078 48.DEF.

A is the hundredths digit of sin 58.23° ___

B is the hundredths digit of the angle of which the secant is 3.10392 ___

C is the tenths digit of the tangent of the angle the cotangent of which is 1.2985 ___

D is the ones digit of the cosecant of the angle of which the cosine is 0.9822 ___

E is the thousandths digit of the secant of the angle the cosine of which is 0.6388 ___

F is the tenths digit of the cosecant of 72.864° ___

If your answer coincides with the stated co-ordinates given at the beginning of the cache page, you’re good to go; otherwise, there is some "sorting" to be done!!

**When ready . . . go for it!**

**The area around Stage Two can accumulate deep snow in the winter and hence, a lot of water in the spring. (We know . . . been there, done that!!) Choose your season and/or take someone with you.**

**
Additional Hints**
(Decrypt)

Bar - fuva zvpeb; gjb - purfg zvpeb; guerr - jnvfg zvpeb; sbhe - ybj erthyne.

Decryption Key

A|B|C|D|E|F|G|H|I|J|K|L|M

-------------------------

N|O|P|Q|R|S|T|U|V|W|X|Y|Z

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