reciprocal

Examples of reciprocal in the following topics:

• Secant and the Trigonometric Cofunctions

  • Trigonometric functions have reciprocals that can be calculated using the unit circle.
  • Each of these functions has a reciprocal function, which is defined by the reciprocal of the ratio for the original trigonometric function.
  • Note that reciprocal functions differ from inverse functions.
  • The cosecant function is the reciprocal of the sine function, and is abbreviated as$\csc$.
  • The other reciprocal functions can be solved in a similar manner.

• Parallel and Perpendicular Lines

  • For two lines in a 2D plane to be perpendicular, their slopes must be negative reciprocals of one another, or the product of their slopes must equal $-1$.  
  • Since $3$ is the negative reciprocal of $-\frac{1}{3}$, the two lines are perpendicular.
  • Again, start with the slope-intercept form and substitute the values, except the value for the slope will be the negative reciprocal.  
  • The negative reciprocal of $\frac{1}{4}$ is $-4$.  
  • The values of their slopes are negative reciprocals of each other; therefore, the angle of intersection is $90$ degrees.

• Complex Fractions

  • From previous sections, we know that dividing by a fraction is the same as multiplying by the reciprocal of that fraction.
  • Therefore, we use the cancellation method to simplify the numbers as much as possible, and then we multiply by the simplified reciprocal of the divisor, or denominator, fraction:
  • Recall, again, that dividing by a fraction is the same as multiplying by the reciprocal of that fraction:

• Simplifying, Multiplying, and Dividing Rational Expressions

  • Recall the rule for dividing fractions: the dividend is multiplied by the reciprocal of the divisor.
  • The same applies to dividing rational expressions; the first expression is multiplied by the reciprocal of the second.
  • Rather than divide the expressions, we multiply $\displaystyle \frac {x+1}{x-1}$ by the reciprocal of $\displaystyle \frac {x+2}{x+3}$:

• Complex Conjugates and Division

  • The reciprocal of a nonzero complex number $z = x + yi$ is given by

• The Law of Sines

• Fractions

  • The process for dividing a number by a fraction entails multiplying the number by the fraction's reciprocal.
  • The reciprocal is simply the fraction turned upside down such that the numerator and denominator switch places.

• The Order of Operations

  • Since multiplication and division are of equal precedence, it may be helpful to think of dividing by a number as multiplying by the reciprocal of that number.

• Introduction to Hyperbolas

• Inverse Trigonometric Functions

  • The reciprocal function is $\displaystyle{\frac{1}{\sin x}}$, which is not the same as the inverse function.