What is the difference between INVERSE and RECIPROCAL?

[tomcenjerrym]

Hi again everyone…

I have 3 questions:

1) What is the difference between INVERSE and RECIPROCAL?

2) Does sin^−1 (u/a) equal to arc sin (u/a)?

3) What is the EASIEST way to remember that integral du / (√(a^2 − u^2)) = sin^−1 (u/a) + C or any transformation of trigonometric integration?

Thanks

 

[jasmin_123]

3 questions, help me

1) https://www.answers.com

RECIPROCAL

adj.

1. Concerning each of two or more persons or things.
2. Interchanged, given, or owed to each other: reciprocal agreements to abolish customs duties; a reciprocal invitation to lunch.
3. Performed, experienced, or felt by both sides: reciprocal respect.
4. Interchangeable; complementary: reciprocal electric outlets.
5. Grammar. Expressing mutual action or relationship. Used of some verbs and compound pronouns.
6. Mathematics. Of or relating to the reciprocal of a quantity.
7. Physiology. Of or relating to a neuromuscular phenomenon in which the excitation of one group of muscles is accompanied by the inhibition of another.
8. Genetics. Of or designating a pair of crosses in which the male or female parent in one cross is of the same genotype or phenotype as the complementary female or male parent in the other cross.

n.

1. Something that is reciprocal to something else.
2. Mathematics. A number related to another in such a way that when multiplied together their product is 1. For example, the reciprocal of 7 is 1/7; the reciprocal of 2/3 is 3/2.

[From Latin reciprocus, alternating.]
INVERSE

adj.

1. Reversed in order, nature, or effect.
2. Mathematics. Of or relating to an inverse or an inverse function.
3. Archaic. Turned upside down; inverted.

n. (ĭn'vûrs', ĭn-vûrs')

1. Something that is opposite, as in sequence or character; the reverse.
2. Mathematics. One of a pair of elements in a set whose result under the operation of the set is the identity element, especially:
1. The reciprocal of a designated quantity. Also called multiplicative inverse.
2. The negative of a designated quantity. Also called additive inverse.

[Middle English, from Latin inversus, past participle of invertere, to invert. See invert.]

 

[pmonon]

Re: 3 questions, help me

Hi again everyone…

I have 3 questions:

1) What is the difference between INVERSE and RECIPROCAL?

2) Does sin^−1 (u/a) equal to arc sin (u/a)?

3) What is the EASIEST way to remember that integral du / (√(a^2 − u^2)) = sin^−1 (u/a) + C or any transformation of trigonometric integration?

Thanks

1) Inverse and reciprocal are different but in special case can be the same. For example, -a is the additive inverse of a, 1/a is the multiplicative inverse of a. 1/a is the reciprocal of a and vice versa and also, one is the multiplicative inverse of the other.

2) Yes, if you dont mean the exponent by your notation of the first term.
3)I think if you remember that (sinΘ)^2 +(cosΘ)^2 and use substitution method you can easily derive those.

 

[harsha_jois]

Re: 3 questions, help me

Hi
First q:
Reciprocal is usually taken as decimal inverse. or 1/ n.
Inverse on the other hand is a function. Sine inverse will give you the angle.

Second q:
If you define the arc causing the sine by θ then they are same.

Third q:
Two-D coordinate geometry would help you remember it.
Integral is the area under the curve.