Trigonometry: Meaning - Inverse Trigonometric Functions
This is a free lesson from our course in Trigonometry
This lesson defines and introduces the properties of the inverse trigonometric functions. The special feature requires inter-connections of the concepts with those previously learnt algebraic principles and techniques. When you'll go through this interactive tutorial you'll explore by analyzing the graph of the function and graph of its inverse. The domain and range of each of the inverse functions is also explored. The Inverse Trigonometric Functions are meant to be the `inverses' of the standard trigonometric functions. The inverse functions of sin, cos and tan are called arcsin, arccos and arctan. The notations sin-1(x), cos-1(x) and tan-1(x) are also used commonly. If y = sin x then x = arcsin y. If y = cos x then x = arccos y. And if y = tan x then x = arctan y. That's what we would like to imply by inverse trig functions, but the definitions technically aren't perfect, simply because the trigonometric functions are not one-to-one. (More text below video...)
<h2> Trigonometry - Meaning - Inverse Trigonometric Functions</h2> <p> Meaning - Inverse Trigonometric Functions</p> <p> An equation for the inverse of y = f(x) = sin x is obtained by interchanging x and y. </p>
Other useful lessons:
Evaluating Inverse Trigonometric Functions
Solving Trigonometric Equations
(Continued from above)           An equation for the inverse of y = f(x) = sin x is obtained by interchanging x and y. The inverse sine of x is
          y = f-1(x) = sin-1 x which means x = sin y where -1 <= x <= 1 and -/2 <= y = /2.           The inverse cosine function of y = f(x) = cos x is y = f-1x = cos-1 x where 0 <= x <=  .           The inverse cosine function of y = f(x) = tan x is y = f-1 x = tan-1 x or y = arctan x where  - <= x <= 
          and -/2 <= y = /2.

More commonly the inverse functions are denoted by:

  • f(x) = arcsin x or f(x) = sin-1 x.
  • f(x) = arccos x or f(x) = cos-1 x.
  • f(x) = arctan x or f(x) = tan-1 x.
  • Definition and the domain of other inverse trigonometric functions:

         y = sec-1 x means x = sec y where |x| and 0 <= y <= ,y  /2.

         y = csc-1 x means x = csc y where |x| and -/2 <= y <= /2, y  0.

         y = cot-1 x means x = cot y where |x| and - < |x|  and  0 <= y <= .

For example, sin-1 1/2 = /6 because sin /6 is 1/2. For the graph of sin-1 , the values of are taken along the x-axis and the vertical y-axis and the graph is obtained as the sine wave turns sideways so that becomes y and y becomes and the function returns to the value between -/2 and /2. Once you go through the instructor's explanation in the video, it'll be easy for you to understand about these inverse trigonometric functions.

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