RPN Scientific 15 manual · Chapter 5 of 12

Functions

Roots, powers, logarithms, trigonometry in three angle units, hyperbolics, conversions, factorial, permutations, combinations, integer and fractional parts, rounding and random numbers.

5.1Square root, square, powers and roots#

√x is the square root of X (an error if X is negative). gx²√x squares X. yˣ raises Y to the power X. Any root is a power with a fractional exponent: the cube root of 27 is 27 to the power 1/3.

Example 5.1

Square root of 144

Key in 144 and press √x.
144√x

Example 5.2

Square of 1.5

Key in 1.5 and press g, x².
1.5gx²√x

Example 5.3

Cube root of 27

Key in 27, ENTER, then 3 and 1/x to get 1/3. Press yˣ.
27ENTER31/xyˣ

\sqrt[3]{27}=27^{1/3}=3

Any root is a power: the n-th root of Y is Y to the power 1/n.

5.2Exponentials and logarithms#

eˣ is e to the X (e is about 2.71828). gLNeˣ is the natural logarithm LN. 10ˣ is 10 to the X. gLOG10ˣ is LOG, the logarithm base 10. LN and LOG need X greater than zero.

Example 5.4

e to the power 1

Key in 1 and press eˣ.
1eˣ

Example 5.5

Natural log of 5

Key in 5 and press g, LN.
5gLNeˣ

Example 5.6

Log base 10 of 1000

Key in 1000 and press g, LOG.
1000gLOG10ˣ

Example 5.7

10 to the power 2

Key in 2 and press 10ˣ.
210ˣ

To find a log in another base, divide logs: the logarithm of 8 to base 2 is LN 8 ÷ LN 2.

Example 5.8

Log of 8 base 2

Take LN of 8, then LN of 2, then divide.
8gLNeˣ2gLNeˣ÷

\log_{2}8=\frac{\ln 8}{\ln 2}=3

Change of base: any logarithm can be found from natural logarithms.

5.3Sine, cosine and tangent#

SIN, COS and TAN work in the current angle unit (degrees unless you change it). The inverse functions are on the g layer: gSIN⁻¹SIN, gCOS⁻¹COS and gTAN⁻¹TAN. Inverse sine and cosine need X between −1 and 1.

Example 5.9

Sine of 30 degrees

Key in 30 and press SIN.
30SIN

Example 5.10

Angle whose sine is 0.5

Key in 0.5 and press g, SIN⁻¹.
0.5gSIN⁻¹SIN

Example 5.11

Tangent of 45 degrees

Key in 45 and press TAN.
45TAN

Example 5.12

Arctangent of 1 in radians

Switch to radian mode with g, 8. Key in 1 and press g, TAN⁻¹.
gRAD81gTAN⁻¹TAN

Example 5.13

Sine of π/4 in radians

Switch to RAD, press g π, key in 4, ÷ and SIN.
gRAD8gπEEX4÷SIN

Radians mode: the RAD indicator shows the angle unit. The display shows: 0.0000 / 0.0000 / 0.0000 / 0.0000. — Radians mode: the RAD indicator shows the angle unit.

\text{angle in degrees}=\text{angle in radians}\times\frac{180}{\pi}

How the angle units relate: 360 degrees = 2π radians = 400 gradians.

5.4Hyperbolic functions#

Press fHYPGTO (HYP) and then SIN, COS or TAN for the hyperbolic function. Press gHYP⁻¹GTO (HYP⁻¹) and then one of the same keys for the inverse. Hyperbolic functions do not depend on the angle unit.

Example 5.14

Hyperbolic sine of 1

Key in 1, press f, HYP, then SIN.
1fHYPGTOSIN

Example 5.15

Hyperbolic tangent of 1

Key in 1, press f, HYP, then TAN.
1fHYPGTOTAN

Example 5.16

Go there and back

Take the hyperbolic sine of 1, then the inverse.
1fHYPGTOSINgHYP⁻¹GTOSIN

HYP⁻¹ COS needs X of at least 1. HYP⁻¹ TAN needs X between −1 and 1. Outside those ranges you see Error 0.

\sinh x=\frac{e^{x}-e^{-x}}{2}\qquad \cosh x=\frac{e^{x}+e^{-x}}{2}\qquad \tanh x=\frac{\sinh x}{\cosh x}

fHYPGTO
HYP
SIN

Hyperbolic sine of the number in X.

5.5The value of pi#

gπEEX puts π (3.14159265359) on the stack.

Example 5.17

Area of a circle of radius 2

Key in 2, press g x², then g π and ×.
2gx²√xgπEEX×

5.6Convert between polar and rectangular coordinates#

g→P1 (→P) turns a point (x, y) into polar form. Put y in Y and x in X. The result is r in X and the angle θ in Y. f→R1 (→R) goes the other way: put θ in Y and r in X, and get x in X and y in Y. The angle uses the current angle unit.

Example 5.18

Rectangular to polar: (3, 4)

Key in y = 3, ENTER, then x = 4, and press g, →P.
3ENTER4g→P1

Example 5.19

Polar to rectangular: r = 5, θ = 53.1301

Key in the angle first, ENTER, then r, and press f, →R.
53.1301ENTER5f→R1

r=\sqrt{x^{2}+y^{2}}\qquad \theta=\arctan\frac{y}{x}\qquad x=r\cos\theta\qquad y=r\sin\theta

The conversions that →P and →R perform.

TZYX

3 ENTER 4

g→P1

36.86995

→P on the point (x, y) = (4, 3): put y in Y and x in X.

5.7Convert degrees, radians and hours-minutes-seconds#

f→RAD3 (→RAD) converts X from degrees to radians. g→DEG3 (→DEG) converts radians to degrees. f→H.MS2 (→H.MS) converts decimal hours (or degrees) to hours.minutes-seconds, written H.MMSS. g→H2 (→H) does the reverse.

Example 5.20

180 degrees in radians

Key in 180 and press f, →RAD.
180f→RAD3

Example 5.21

π radians in degrees

Press g π, then g →DEG.
gπEEXg→DEG3

Example 5.22

2.5 hours as hours.minutes

Key in 2.5 and press f, →H.MS.
2.5f→H.MS2

Example 5.23

1 hour 30 minutes as decimal hours

Key in 1.3 and press g, →H.
1.3g→H2

180^{\circ}=\pi\ \text{rad}\qquad 2.5\ \text{h}\to 2\text{ h }30\text{ min}\ (2.3000)

5.8Factorial#

fx!0 (x!) gives the factorial of X. It also works for non-whole numbers (it uses the gamma function). Negative whole numbers give Error 0. Large values overflow to 9.999999999 × 10⁹⁹.

Example 5.24

Factorial of 5

Key in 5 and press f, x!.
5fx!0

Example 5.25

Factorial of one half

Key in 0.5 and press f, x!.
0.5fx!0

n!=n(n-1)(n-2)\cdots 1\qquad x!=\Gamma(x+1)

Factorial for whole numbers, and its extension to other numbers.

5.9Permutations and combinations#

fPy,x+ (Py,x) counts ordered selections of X items from Y items. gCy,x+ (Cy,x) counts unordered selections. Both need whole numbers, with X not larger than Y.

Example 5.26

Permutations of 4 items taken 2 at a time

Key in 4, ENTER, 2 and press f, Py,x.
4ENTER2fPy,x+

Example 5.27

Combinations of 52 cards taken 5 at a time

Key in 52, ENTER, 5 and press g, Cy,x.
52ENTER5gCy,x+

{}_{n}P_{r}=\frac{n!}{(n-r)!}\qquad {}_{n}C_{r}=\frac{n!}{r!\,(n-r)!}

Py,x and Cy,x with n = Y and r = X.

5.10Integer part, fractional part and absolute value#

gINTSTO (INT) keeps the whole part of X. fFRACSTO (FRAC) keeps the part after the point. gABSCHS (ABS) removes a minus sign. For a negative number, INT and FRAC both keep the minus sign.

Example 5.28

Integer part of 2.75

Key in 2.75 and press g, INT.
2.75gINTSTO

Example 5.29

Fractional part of 2.75

Key in 2.75 and press f, FRAC.
2.75fFRACSTO

Example 5.30

Absolute value of −4.37

Key in 4.37, CHS, then g, ABS.
4.37CHSgABSCHS

2.75=\underbrace{2}_{\mathrm{INT}}+\underbrace{0.75}_{\mathrm{FRAC}}

5.11Round to the displayed digits#

The display setting only changes what you see. The calculator keeps 10 digits (and works with 13 inside). gRNDx⇄y (RND) really rounds the number in X to the digits shown, which is useful for money.

Example 5.31

Round 1.23456 to two decimals

Switch to FIX 2, key in 1.23456 and press g, RND. Then switch to FIX 9 to prove it.
fFIX721.23456gRNDx⇄yfFIX79

5.12Random numbers#

fRAN#ENTER (RAN#) puts a random number between 0 and 1 into X, with 10 digits. Each press gives a new number. To make the sequence repeat, store a seed first: key in a number, press STO, then f and ENTER. The same seed always gives the same sequence. RCL then fRAN#ENTER recalls the seed.

To get a whole number from 1 to 6, multiply by 6, add 1 and take the integer part: RAN#, 6, ×, 1, +, INT.

HP-15C is a trademark of HP Inc.. Trident Calculator is an independent product by South View Studios and is not affiliated with, sponsored by or endorsed by HP. The name is used only to describe the kind of calculator this model resembles; Trident's names, keys, display, fonts and software are its own clean-room design.