Use of trig identities

[DaftHuman01]

Hi. Does anyone know how I can use the HP Prime to automatically apply trig identities. I have attached an example of the equation and what the result could be. Thanks!

[C.Ret]

Bonjour,

You are welcome.

This type of linearization reminds me of the time when we used to simplify the applied equations to facilitate computations from abacuses or a slide rule. For this specific case, this linearization also has the advantage of giving the shape of the signal as a function of time.

Today, with the power of smartphones and personal digital assistants, I am not sure that this is still useful. However, the HP Prime can help quite easily with this trigonometric linearization:


It can also easily do the calculation with the first expression containing several trigonometric functions.

[Albert Chan]

(10-01-2024 03:19 PM)C.Ret Wrote:  This type of linearization reminds me of the time when we used to simplify the applied equations to facilitate computations from abacuses or a slide rule.

Product of 2 numbers, cos product to cos sum, vs log/exp

Division of 2 numbers, cos product to cos sum, acos(1/x) = asec(x)

[Albert Chan]

cos product to cos sum identity proof, using complex numbers.

cos(a)*cos(b)
= (cis(a)+cis(-a))/2 * (cis(b)+cis(-b))/2
= (cis(a+b)+cis(-(a+b))/4 + (cis(a-b)+cis(-(a-b))/4
= cos(a+b)/2 + cos(a-b)/2

Cas> simplify(exp2trig(trig2exp(cos(a)*cos(b))))

1/2*cos(a+b) + 1/2*cos(a-b)

[DaftHuman01]

(10-01-2024 03:19 PM)C.Ret Wrote:  Bonjour,

You are welcome.

This type of linearization reminds me of the time when we used to simplify the applied equations to facilitate computations from abacuses or a slide rule. For this specific case, this linearization also has the advantage of giving the shape of the signal as a function of time.

Today, with the power of smartphones and personal digital assistants, I am not sure that this is still useful. However, the HP Prime can help quite easily with this trigonometric linearization:


It can also easily do the calculation with the first expression containing several trigonometric functions.

Thank you for the help. It seems kind of strange though to have to put decimal points after every number for this to work though. Can you also help me with how I can solve the problem in the attached image with the HP Prime as well? Thank you in advance.

[C.Ret]

(10-04-2024 12:27 AM)DaftHuman01 Wrote:  It seems kind of strange though to have to put decimal points after every number for this to work though.

I deliberately put decimal points at each number in order to be consistent.

But, in general these points are present in order to control the type of result that is expected.
Without decimal points, expressions composed of integers will be considered exact by the CAS of the HP Prime and will give exact results even if it means producing extended expressions. The same type of results can be obtained by using the exact command.

The decimal points indicate that it is an approximate value and exempt the CAS system from generating an exact expression. This allows you to obtain a numerical value.

The interest here is to obtain an approximate value of the additive constant as in the formula given as an example (i.e. 344.146).

If we do without the decimal points or if we use the exact command, the expression obtained does not give a numerical approximation but the exact expression of this additive constant (i.e. 600*cos(55°) ).

(10-04-2024 12:27 AM)DaftHuman01 Wrote:  Can you also help me with how I can solve the problem in the attached image with the HP Prime as well?

I am not aware of any way to control the appearance of complex numbers in polar format in the HP Prime CAS. As such, there is no easy and immediately accessible way to achieve the simplification performed in this example.

Unlike other machines, there is no way in CAS mode to choose polar format for complex numbers.

In HOME mode, any complex result can be displayed in cartesian or polar format simply by pressing the shifted toggle key [ Shift ][ ∡ ]. But this functionality does not exist in CAS mode.

This state of affairs has already been noted and has already been discussed by others on the forum, for example here under the input of Dands.



On the other hand, this is not really a simplification by trigonometric properties, but a simple simplification of the numerator and denominator by the some 80 factor.

Indeed, \( 80\angle 36.8699 = 80\cdot \left ( 1\angle 36.8699 \right ) \) as well as \( 80\angle 36.8699 = 64+48\, \textit{i} = 80\cdot \left ( 0.8+0.6\, \textit{i} \right ) \) and \( 160 = 2\cdot 80 \).