Question

Someone please explain to me how to get the answers to these 3 problems. (also ignored the darken circles since they are incorrect)

Answer 1

Answer:

  4.  y is the angle between -π/2 and π/2, inclusive, whose sine equals x

  5.  QI and QII

  6.  QI and QIV

Step-by-step explanation:

4. Just as an exponent of -1 indicates the inverse (reciprocal) of a variable ...

  $x^{-1}=\dfrac{1}{x}$

so an "exponent" of -1 on a function name indicates the inverse of the function. (The same notation is used to indicate the inverse of other things, as well.) That is, the notation ...

  $y=\sin^{-1}{(x)}$

indicates the inverse of the sine function, also called the arcsine function. The above notation can be read as "y equals the angle whose sine is x".

In other words, the sine function takes an angle as an argument and gives a pure number. The arcsine function takes a pure number as an argument and gives an angle.

(Unless otherwise specified, the default measure for angles is radians. Most calculators can be set to use angles in degrees or in radians. You need to be sure which mode you have set.)

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Comment on the graph

The attached graph shows one full cycle of sine, cosine, and tangent functions. Each function has a portion highlighted by dots. The highlighted portion corresponds to the portion for which the inverse function is defined. This is a portion of the curve that passes the horizontal line test, so the result of the inverse function is single-valued as a function result must be.

I have also identified the angle quadrants as Q3, Q4, Q1, Q2. By comparing the dotted function sections to the quadrant identifiers, you can see the answers to problems 5 and 6.

The image of where the various inverse trig functions are defined is a good one to keep in your head whenever you're working with those functions.

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5. The arccos function is defined in quadrants I and II.

6. The arctan function is defined in quadrants IV and I.