HP 35s User Manual

							 
 
hp calculators 
 
 
 
 
HP 35s  Solving Simple Trigonometry Problems 
 
 
 
 
The trigonometric functions 
 
Degrees, radians and gradians 
 
Practice working problems involving trig functions 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
						

							 
hp calculators 
 
HP 35s  Solving Simple Trigonometry Problems 
 
hp calculators - 2 - HP 35s  Solving Simple Trigonometry Problems - Version 1.0 
The trigonometric functions 
 
The trigonometric functions, sine, cosine, tangent, and related functions, are used in geometry, surveying, and design. 
They also occur in the solutions to orbital mechanics, integration, and other advanced applications. 
 
The HP 35s provides the three basic functions, and their inverse, or “arc” functions. These work in degrees, radians and 
gradians modes. In addition, ! is provided as a function on the right-shifted “cos” key. 
 
The secant, cosecant and cotangent functions are easily calculated using the !, , and # keys respectively, 
followed by $. To help remember whether the secant function corresponds to the inverse sine or cosine, it can be 
helpful to note that the first letters of “secant” and “cosecant” are inverted in relation to those of “sine” and “cosine”, just 
as the secant and cosecant are the inverted cosine and sine functions. 
 
Trigonometric modes 
 
The HP 35s can calculate trigonometric functions in any of these three modes: Degrees, Radians or Gradians. 
 
Practice working problems involving trig functions 
 
Example 1: Select the appropriate angle mode. 
 
Solution: Press the 9 key below the screen.%
 
 Figure 1 
 
 Press &,  or ( to select DEGrees, RADians or GRADians mode, or use the arrow keys ), *, 
+ and , to select the required mode and then press -. For example, to select RAD, press .%
 
Answer: The selected trigonometric mode is displayed at the top of the screen if it is RAD or GRAD. If no angle 
mode is shown, then the mode is degrees. The 9 command works the same way in algebraic and in 
RPN modes. 
 
 There are 360 degrees, or 2 ! radians in a circle. Gradians mode divides each quarter of a circle into 100 
parts, in a sort of decimal system, making 400 gradians in a circle. 
 
 It is very easy to forget that one angle mode is set but angles are being entered in a different mode. Making 
it a habit to check the angle mode is a good policy. The commands DEG, RAD and GRAD can be entered 
into programs, and it is worth using them to be sure that a program will work as required. 
 
Example 2: What is the sine of !/2 radians? 
 
Solution: In RPN mode, press: ./0. 
 In algebraic mode, ./0- 
   
						

							 
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HP 35s  Solving Simple Trigonometry Problems 
 
hp calculators - 3 - HP 35s  Solving Simple Trigonometry Problems - Version 1.0 
 Figure 3 
Answer:  The sine of !/2 radians is calculated as exactly 1. Answers will not always be exact as in this case. The  
HP 35s works with 12 decimal digits, so trigonometric calculations can be expected to be accurate to 12 
decimal places. For example the sine of ! radians is calculated as 2!10-13, displayed as 2E-13, which is 
correctly equal to zero to 12 decimal places. 
 
Example 3: Show that the rule sin²(x) + cos²(x) = 1 applies correctly when x is 45°. 
 
Solution: First, remember to set the required angle mode. Press 9&. 
 
 In algebraic mode, work through the problem by calculating the sine, squaring it, then adding the square of 
the cosine: 
 
%1234**512!34-%
%
 Figure 4 
 
 The same can be done in RPN mode. First, calculate the sine and cosine of 45°. Then add: 
 
%34-126!125%
 
 Figure 5 
  
Answer: Both the algebraic and the RPN calculations confirm that the rule sin²(x) + cos²(x) = 1 applies correctly 
when x is 45°. 
 
Example 4: A ladder is leaning against a vertical wall. The ladder is 6 meters long and the foot of the ladder is 3 meters 
from the base of the wall. What is the angle between the top of the ladder and the wall? 
 
Solution: In RPN mode, divide the side opposite the angle by the long side and get the arc sine: 
(-7018 
 
 In algebraic mode, 18(07- 
 
 Figure 6 
 
Answer:  The ladder is at an angle of 30 degrees from the wall. 
   
						

							 
 
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HP 35s  Logarithmic functions 
 
 
 
 
Log and antilog functions 
 
Practice working problems involving logarithms 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
						

							 
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HP 35s  Logarithmic functions 
 
hp calculators - 2 - HP 35s  Logarithmic functions - Version 1.0 
Log and antilog functions 
 
Before calculators like the HP 35s became easily available, logarithms were often used to simply multiplication. They are 
still used in many subjects, to represent large numbers, as the result of integration, and even in number theory. 
 
The HP 35s has four functions for calculations with logarithms. These are the “common” logarithm of “x”, !, its 
inverse, , the “natural” logarithm of “x”, # and its inverse, $. 
 
Common logarithms are also called “log to base 10” and the common logarithm of a number “x” is written 
 
    LOG10 x    or just    LOG x 
 
Natural logarithms are also called “log to base e” and the natural logarithm of a number “x” is written 
 
    LOGe x    or   LN x 
 
, and $ are also called “antilogarithms” or “antilogs”. $ is also called the “exponential” function or “exp”. Apart 
from being the inverses of the log functions, they have their own uses.  is useful for entering powers of 10. $ is 
used in calculations where exponential growth is involved.  
 
Practice working problems involving logarithms 
 
Example 1: Find the common logarithm of 2. 
 
Solution: In RPN mode:  %&!

 Figure 1 
 
 In algebraic mode:  &!%(

 Figure 2 
 
Answer: The common logarithm of 2 is very nearly 0.3010. 
 
Example 2: What is the numeric value of the base of natural logarithms, e? 
 
Solution: This is a quick way to type the value of e. 
 
 In RPN mode:  )*$ 
 
 In algebraic mode: *$)(
   
						

							 
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HP 35s  Logarithmic functions 
 
hp calculators - 3 - HP 35s  Logarithmic functions - Version 1.0 
 Figure 3 
 
Answer:  e is equal to 2.71828182846. The pattern 18 – 28 – 18 – 28 is easy to remember. 
 
Example 3: What is the value of X, in the equation: 2X = 8? 
 
Solution: To solve this example, well apply one of the properties of logarithms which states that the logarithm of an 
base taken to a power is equal to the power multiplied by the log of the base. This involves taking the 
logarithm of both sides of the equation. The original equation would then look like this: 
 
X LOG(2) = LOG(8)  Figure 4 
      
 X is therefore equal to: 
 
)2(
)8(
LOG
LOGX!   Figure 5 
 
 In RPN mode:  +&!%&!,

 In algebraic mode:  &!+-,&!%(

 Figure 6 
 
Answer: The value of X is 3. Figure 6 shows the result in algebraic mode. Note that the same answer will be found 
using natural logarithms or common logarithms.   
						

							 
 
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HP 35s  Hyperbolic functions 
 
 
 
 
Hyperbolic trigonometric functions 
 
Practice using hyperbolic trigonometric functions 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
						

							 
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HP 35s  Hyperbolic functions 
 
hp calculators - 2 - HP 35s  Hyperbolic functions - Version 1.0 
Hyperbolic trigonometric functions 
 
Trigonometric functions are often called “circular” functions, because the value for the cosine and sine of an angle lie on 
the unit circle defined by X2 + Y2 = 1 (points on the unit circle will have the X and Y coordinate of (Cosine(theta), 
Sine(theta))). Hyperbolic trigonometric functions have a similar relationship, but with the hyperbola defined by the 
equation X2 – Y2 = 1. 
 
Given a value for Z, the hyperbolic sine is calculated by evaluating the following: 
  Figure 1 
 
The hyperbolic cosine is calculated by evaluating the following: 
  Figure 2 
 
Assume that Z is 3. The position on the unit hyperbola X2 – Y2 = 1 is defined by the point (COSH(Z),SINH(Z)), where 
COSH is the hyperbolic cosine and SINH is the hyperbolic sine. The value for the SINH(3) is equal to 10.0179 and the 
value of COSH(3) is 10.0677. When 10.0677 x 10.0677 – 10.0179 x 10.0179 is evaluated, the value is 1, so the point 
falls on the unit hyperbola. The hyperbolic tangent is defined as the hyperbolic sine divided by the hyperbolic cosine. 
 
Hyperbolic functions have applications in many areas of engineering. For example, the shape formed by a wire freely 
hanging between two points ( known as a catenary curve) is described by the hyperbolic cosine (COSH). Hyperbolic 
functions are also used in electrical engineering applications. 
 
On the HP 35s, hyperbolic functions are access by pressing the ! keys and then the appropriate trigonometric 
key or inverse trigonometric key. 
 
Practice using hyperbolic trigonometric functions 
 
Example 1: Find the Hyperbolic Sine of 2. 
 
Solution: In RPN mode, #!$%
 In algebraic mode, !$#! 
%
 Figure 3  
Answer: 3.62686. Figure 3 shows the display in algebraic mode. 
 
Example 2: A tram carries tourists between two peaks that are the same height and 437 meters apart. Before the tram 
 latches onto the cable, the angle from the horizontal to the cable at its point of attachment is 63 degrees. 
 How long does it take the tram to travel from one peak to the other, if the tram moves at 135 meters per  
 minute?