HP 33S User manual

hp calculators
HP 33S Solving Simple Trigonometry Problems
The trigonometric functions
Degrees, radians and gradians
Practice working problems involving trig functions

hp calculators
HP 33S Solving Simple Trigonometry Problems
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 33S provides the three basic functions, and their inverse, or “arc” functions. These work in degrees, radians and
gradians modes. In addition, πis provided as a function on the right-shifted “cos” key.
The secant, cosecant and cotangent functions are easily calculated using the n, k, and qkeys 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 33S 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 Ýkey below the screen.
Figure 1
Press 1, 2or 3to select DEGrees, RADians or GRADians mode, or use the arrow keys Ö,
Õ, ×and Øto select the required mode and then press Ï. For example, to select RAD,
press 2.
Figure 2
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 Ýcommand works the same way in algebraic and in
RPN modes.
There are 360 degrees, or 2 πradians 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.
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hp calculators
HP 33S Solving Simple Trigonometry Problems
Example 2: What is the sine of π/2 radians?
Solution: In RPN mode, calculate π/2, then press k.
ºj2¯k.
In algebraic mode calculate π/2 in parentheses, then press k.
ºyºj¯2º|k. Following this with Ïis not necessary, but it completes the
calculation by putting an equals sign at the end.
Figure 3
Answer: The sine of π/2 radians is calculated as exactly 1. Answers will not always be exact as in this case. The
HP 33S works with 12 decimal digits, so trigonometric calculations can be expected to be accurate to 12
decimal places. For example the sine of πradians 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 Ý1.
In algebraic mode, work through the problem by calculating the sine, squaring it, then adding the square of
the cosine:
45k=Ù45n=Ï
Figure 4
The same can be done in RPN mode. First, calculate the sine and cosine of 45°. Then add:
45Ïk=wn=Ù
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°.
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hp calculators
HP 33S Solving Simple Trigonometry Problems
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:
3Ï6¯¹i
In algebraic mode, calculate 3/6 in parentheses, then the arc sine:
ºy3¯6º|¹i
Following this with Ïis not necessary, but it completes the calculation by putting an equals sign at
the end.
Figure 6
Answer: The ladder is at an angle of 30 degrees from the wall.
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