4.7 Inverse Trigonometric Functions

For a function to have an inverse, it must pass the Horizontal Line Test. Some trigonometric functions do not pass the Horizontal Line Test on their own and thus must be restricted.

Inverse Sine Function



As you can see the graph of y=sin x does not pass the horizontal line test on its own.








When y=sin x restricts the domain to the interval [-π/2, π/2] it passes the horizontal line test and can now have an inverse function.







The inverse sine function is represented by y=arcsin x or y= sin^-1 x

Remember: the sin^-1 x is the angle (or number) whose sine is x











Evaluating the Inverse Sine Function

So lets say you have a problem like this:
sin^-1(-1/2)

You are trying to find the angle whose sine is -1/2

If you look at the Unit Circle you will notice that at -π/6 the sin is -1/2

sin^-1(-1/2)= -π/6



Inverse Cosine Function


In the cosine function, the domain is restricted to the interval [0,π] to make it 1-1 (pass the horizontal line test).








The inverse cosine function is represented by y=arccos x or y= cos^-1 x

Remember: the cos^-1 x is the angle (or number) whose cos is x











Evaluating the Inverse Cosine Function

You evaluate the inverse of cosine functions the exact same way as you do with inverse of sine functions.

Ex: arccos(-1)

Angle whose cosine is -1?? π

arccos(-1)= π


Inverse Tan Function



In the tangent function, the domain is restricted to the interval (-π/2, π/2) to make it 1-1 (pass the horizontal line test).













The inverse tangent function is represented by y=arctan x or y= tan^-1 x

Remember: the tan^-1 x is the angle (or number) whose tan is x











Evaluating the Inverse Tangent Function

You evaluate the inverse of tangent functions the exact same way as you do with inverse of sine and cosine functions.



ALSO remember that the domain and ranges of the inverse functions are reverse from the parent function

EX:
Sin x
D: [-π/2, π/2]
R: [-1, 1]

Sin^-1 x
D: [-1, 1]
R: [-π/2, π/2]



Lastly remember that when you see:

cos(arccos x)

the cos and arccos cancel out making

cos(arccos x) = x

*This is the same for sine and tangent as well

Identities

Identity- An equation that is always true

Ex: 2x+5 = 2(x-3) + 11 <---- No matter what you put in for X the sides will always be equal

Identities are used for substitution

Reciprocal Identities:

Quotient Identities:






Pythagorean Identities:

Pythagorean Identities can be derived from other equations. Such as-









Using those two factors we can derive the first pythagorean identity by using the pythagoean theorem and finding a new identity that we can use to substitute.

Plug in the factors

distribute the square

factor out a c^2

divide by c^2 to get the first Pythagorean identity

The other two identities can be derived in a similar way. The three Pythagorean identities are:

Chapter 4, Section 1

Chapter 4, Section 1
Radian Degree and Measure

• The word trigonometry is a word derived from Greek language meaning "measurement of triangles." It deals with the relationships among the sides and angles of triangles
  
  
• An angle is determined by rotating a ray about its endpoint
  

• The starting position of a ray is the initial side of the angle, and the position of the ray after rotation is the terminal side (shown below)
  
  


• Positive angles are generated by a counterclockwise rotation
• Negative angles are genereated by a clockwise rotation

Radian Measure

• One radian is the measure of a central angle (theta) that intercepts an arc (s) equal in length to the radius (r) of the circle


• Because the circumference of a circle is 2PIEr, it follows that a central angles of one full counterclockwise revolution corresponds to an arc length of s = 2PIEr. Therefore, 2PIE radians corresponds to 360, PIE radians corresponds to 180, and PIE over 2 radians corresponds to 90. Because 2PIE = about 6.28, there are just over six radius lengths in a full circle.
  
• In general, the radian measure of a central angle theta is obtained by dividing the arc length (s) by r. Therefore, s/r= theta, where theta is measured in radians.
  




• Because the radian measure of an angle of one full revolution is 2PIE, you can reason the following:

1/2 revolution = 2PIE/2 = PIE radians

1/4 revolution = 2PIE/4 = PIE/2 radians

1/6 revolution = 2PIE/6 = PIE/3 radians

• There are four quadrants in a coordinate system and are numbered I, II, III, and IV

Coterminal Angles

• Two angles are coterminal if they have the same initial and terminal sides

Example: The angles 0 and 2PIE are coterminal, as are the angles PIE/6 and 13PIE/6

• You can find that an angle is coterminal to a given angle theta by adding or subtracting 2PIE (one revolution)

Example: For the positive angle 13PIE/6, subtract 2PIE to obtain a coterminal angle

13PIE/6 - 2PIE = PIE/6

Degree Measure

• A second way to measure angles is in terms of degrees
  
• A measure of one degree is equivalent to a rotation of 1/360 of a complete revolution about the vertex

Conversions Between Degrees and Radians

1. To convert degrees to radians, multiply degrees by PIE radians/180 degrees
   
2. To convert radians to degrees, multiply radians by 180 degrees/PIE radians

*PIE radians = 180 degrees

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Chapter 4, Section 2

Trigonometric Functions: The Unit Circle

4.2: Trigonometry

Okay, so when theres like a right triangle, its got 6 different trigonometric things, these are called:

Sin (oposite leg / hypotenuse)

Cosine (adjacent leg / hypotenuse)

Tangent (opposite leg / adjacent leg)
(also simply sin / cos)

Each of these has a reciprocal:

cosecant = 1 / sin

Secant = 1 / cosine

Cotangent = 1 / tangent

These are reciprocals NOT inverses

On the unit circle, the sin = y value and the cos = x value

a trick to remeber: every 15 degrees = pie / 12

examples:

• 30 degrees = 2 pi / 12 = pi / 6
• 45 degrees = 3 pi / 12 = pi / 4
• 90 degrees = 6 pi / 12 = pi / 2

2.6: Rational Functions

Rational Functions


A rational function is a function that can be written as

Where N(x) and D(x) are polynomials.

• Example of a rational function:

A function likeis considered a rational function, because it can be rewritten as

*NOTE: When you divide by a fraction, the answer gets larger.

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Vertical Asymptotes

Consider the equation .

The denominator cannot be 0, so x ≠ 3.

 Notice that the graph avoids x = 3, and



There can be more than one vertical asymptote, as is demonstrated in the equation, because x = ± 1.

 

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Horizontal Asymptotes

Unlike vertical asymptotes, there can only be one horizontal asymptote.

In order to determine what the horizontal asymptote is

• Pretend that x is a large number like 1,000,000,000
• Consider the highest degree in the numerator and denominator

Because x is so large, everything but the highest degree polynomial is irrelevant. There are three different horizontal asymptote scenarios.

1. Higher degree polynomial in the denominator

 

 Look only at the highest degree polynomials, 3x and 2x²

, so the H.A. is 0.

2. Highest degree polynomial same in both numerator and denominator

Look at only the highest degree polynomials, 3x² and 2x². The x²s cancel each other out, and , so the H.A. is 3/2.

3. Higher degree polynomial in the numerator

It does not level off, so and there is NO H.A.

 Still confused? CLICK HERE to watch a video explaining rational function asymptotes.