How Far to Tokyo?

Everything inside the acos is in degrees, so do that part in DEG. But the acos itself comes back as an angle, and the arc-length formula d = R · θ needs that angle in radians.

The tidy trick on an RPN machine: leave the calculator in DEG for all the sines and cosines, then switch it to RAD just before taking the arccos. The arccos then comes out in radians already and you can multiply by 6371 directly — no × π ÷ 180 step at all.

Spherical law of cosines, giving the central angle θ between the two points:

cos(θ) = sin(lat1)·sin(lat2) + cos(lat1)·cos(lat2)·cos(lon2 − lon1)

Arc length from the central angle (θ must be in radians here):

d = R · θ

Answer: 9,559 km

Set DEG (MODES menu → DEG), then:

51.5074  COS
35.6895  COS
×                        cos(lat1)·cos(lat2)
139.6917  ENTER  0.1278  +      the longitude difference, 139.8195
COS
×
51.5074  SIN
35.6895  SIN
×
+                        the whole argument of the arccos

Now switch to RAD (MODES menu → RAD) and finish:

ACOS                     85.9635775…° becomes 1.50034746… rad
6371  ×

Display: 9558.7137

The angle mode is the physical Р-ГРД-Г slide switch. Start with it on Г (degrees). sin is F 7, cos is F 8, arccos is F 5.

51.5074  F 8
35.6895  F 8
×
139.8195  F 8
×
51.5074  F 7
35.6895  F 7
×
+

Now slide the switch to Р (radians), then:

F 5                      arccos, in radians
6371  ×

Display: 9558.7132

The MK-61 carries eight significant digits, so its last digit sits a little off the HP-42S’s 9558.7137. Both round to 9559 km.

Identical method. On the 15C, DEG is g 7, RAD is g 8, and COS⁻¹ is g COS.

g 7                      DEG
51.5074  COS
35.6895  COS
×
139.6917  ENTER  0.1278  +
COS
×
51.5074  SIN
35.6895  SIN
×
+
g 8                      RAD
g COS                    COS⁻¹, in radians
6371  ×

Result: ≈ 9558.7137 (in f 4, four decimal places) → 9559 km.

ASIN/ACOS/ATAN are built in on every 41C — no module needed.

DEG
51.5074  COS
35.6895  COS
×
139.6917  ENTER  0.1278  +
COS
×
51.5074  SIN
35.6895  SIN
×
+
RAD
ACOS                     85.9635775…° becomes 1.50034746… rad
6371  ×

Display (in FIX 4): 9558.7137 → 9559 km.

RPL keeps RPN’s stack, ENTER, and postfix functions, so the same keystroke sequence as the 41C/42S works unchanged. SIN and COS are unshifted keys and ACOS is ← COS; DEG and RAD are not keys at all on this machine — type them as command names on the alpha keyboard (α before each letter), or set the mode from the MODES menu.

One genuine RPL difference worth knowing: you can also key the whole formula in as a single algebraic object (put it in ' ' quotes and hit EVAL) instead of stepping through the stack — but that object evaluates in one pass under whatever the current angle mode is at that moment, so it can’t switch DEG mid-expression to RAD for just the ACOS step. The mode-switching trick that avoids an explicit × π ÷ 180 only works with sequential stack entry:

DEG
51.5074  COS
35.6895  COS
×
139.6917  ENTER  0.1278  +
COS
×
51.5074  SIN
35.6895  SIN
×
+
RAD
ACOS
6371  ×

Result: 9558.71369499 → 9559 km, matching the HP-42S value to eleven significant digits; the tail is the 48’s real twelve-digit BCD arithmetic.

The TI is algebraic, so the mode-switching trick doesn’t fit as neatly; stay in Degree mode (MODE → Degree) and convert at the end. cos⁻¹ is 2nd COS, and π is 2nd ^.

6371cos⁻¹(sin(51.5074)sin(35.6895)+cos(51.5074)cos(35.6895)cos(139.8195))π/180

Result: ≈ 9558.7137 → 9559 km.

Same single expression as the TI-84. Set Degree mode first: on a ClassWiz (fx-991EX/CW) that is SHIFT MENU → Angle Unit → Degree; on an older fx-991ES it is SHIFT MODE → Degree.

6371 × cos⁻¹(sin(51.5074)sin(35.6895)+cos(51.5074)cos(35.6895)cos(139.8195)) × π ÷ 180

Result: ≈ 9558.7137 → 9559 km.