Chapter 5… Inverse Trigonometric Functions…Section 5.1

Trigonometry

5.1 INVERSE TRIGONOMETRIC FUNCTIONS A relation between x and y is a function if each x-value corresponds to only one y-value. A relation is a function if it passes the Vertical Line Test. For a function to have an inverse, it must be one-to-one and pass the Horizontal Line Test. Function Notation: Inverse Notation: The graph of an inverse function is determined by reversing the coordinates of each ordered pair on the graph of the original function. Also, the domain and range intervals will be switched from the original function to its inverse. Inverse Sine Function The graph of the sine function does not pass the Horizontal Line Test, however, by restricting the domain to the interval  2  x   2 , then we are able to define an inverse.  y  sin x Notice that the interval selected to define sine inverse is increasing. Likewise, the 1/ graph of arcsine will also increase. Definition of Inverse Sine Function The inverse sine function is defined by y  arcsin x and y  sin 1 x if and only if where 1  x  1 and  2  y   2 .

x -/2

/2 -

sin y  x -1

The domain of y  arcsin x is [-1, 1] and the range is [-/2, /2]. Note: The notation sin 1 x  1 sin x . The arcsine of x comes from the association of a central angle with its intercepted arc length on a unit circle. "The arcsine of x is the angle (or number) whose sine is x." Example #1: Graph the arcsine of x using a table of values from the sine function in the interval [-/2, /2]. Y x=sin y Domain =

Range =

Inverse Cosine Function By restricting the domain of the cosine function to the interval 0  x   , then we are able to define an inverse. Notice that the cosine function is decreasing on this interval. So the arccosine y graph will also decrease. y  cos x 1 Definition of Inverse Cosine Function The inverse cosine function is defined by y  arccos x and y  cos 1 x if and only if cos y  x where 1  x  1 and 0  y   . The domain of y  arccos x is [-1, 1] and the range is [0, ].

x 0 -1

Example #2: Graph the arccosine of x using a table of values from the cosine function in the interval [0, ]. y x=cos y Domain =

Range =

1



Chapter 5… Inverse Trigonometric Functions…Section 5.1

Trigonometry

Inverse Tangent Function The inverse of the tangent function is defined within one period of the tangent function on the interval  2  x   2 . The vertical asymptotes of the tangent function become horizontal asymptotes in the inverse function. Definition of Inverse Tangent Function The inverse tangent function is defined by y  arctan x and y  tan 1 x if and only if tan y  x where   x   and  2  y   2 . The domain of y  arccos x is (-, ) and the range is (-/2, /2). Example #3: Graph the tangent function over one period,  2  x   2 . Plot 3 ordered pairs on this function and label the vertical asymptotes. Then, graph the arctangent function by reversing the x and y-values of the coordinates on the tangent function. Be sure to label the horizontal asymptotes of the arctangent function.

Composition of Functions Recall that inverse functions have the property: f ( f 1 ( x ))  x and f 1 ( f ( x ))  x . Inverse Identities for Trigonometric Functions If 1  x  1 and  2  y   2 , then sin (arcsin x) = x and arcsin (sin y) = y If 1  x  1 and 0  y   , then cos (arccos x) = x and

arccos (cos y) = y

If x is a real number and  2  y   2 , then tan (arctan x) = x and arctan (tan y) = y

Example #4: Using a calculator, evaluate each expression to 4 decimal places. Be sure your calculator is in RADIAN MODE. a) sin 1 (0.2903) b) arccos( 0.8114) c) arcsin( 2.305) d) tan 1( 25.45)

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Chapter 5… Inverse Trigonometric Functions…Section 5.1

Trigonometry

Example #5: Evaluate each expression without using a calculator. First solve each expression in terms of sin y, cos y or tan y then evaluate the appropriate radian angle measure. 1 a) arcsin   b) sin 1  1 2

 3 c) arccos    2   

e) arctan



d) cos cos 1 (0.4)

 3

f) arcsec



 2

Other Inverse Trigonometric Functions The remaining inverse trigonometric functions also have defined restricted domains. These restricted values help to determine exact values for the inverse functions. We will not be graphing the remaining inverse functions. Definition of Inverse Trigonometric Functions Function

Domain

Range

Defined Quadrants

y  arcsin x if and only if sin y  x

1  x  1

 2  y   2

Q I, IV

y  arccos x if and only if cos y  x

y  arctan x if and only if tan y  x y  arc cot x if and only if cot y  x y  arc csc x if and only if csc y  x y  arc sec x if and only if sec y  x

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Chapter 5… Inverse Trigonometric Functions…Section 5.1

Trigonometry

Example #6: Using a calculator, evaluate each expression to 4 decimal places, if possible. Be sure your calculator is in RADIAN MODE. a) sin  tan 1(3.4)  b) sec sin 1 ( 0.3446)  c) cot  arccos( 0.5036)  d) csc arcsin 2

Example #7: Find the exact values of the expression without a calculator. Make a sketch of a right triangle.    1   3  a) csc  cos 1   b) sin cos 1      5   3   

  1  c) tan sin 1    5   

  1  d) sec  arc cot    3   

e) tan  arccos x 

 x  f) csc  arctan  7 

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Chapter 5… Inverse Trigonometric Functions…Section 5.2

Trigonometry

1

Chapter 5… Inverse Trigonometric Functions…Section 5.3

Trigonometry

5.3 SOLVING TRIGONOMETRIC EQUATIONS Conditional equations are equations that are true for SOME values, but false for other values. For example sin x = 0 is true for x = 0, , 2, … and for all values: x = 0 ± k = ±k. Note: Recall that x is in terms of Radians and  is in terms of Degrees. Example #1: Determine for which values of  make cos  = -½.

Guidelines for Solving Trigonometric Equations 1. Solve for a trig function first [sin x = or cos x = or tan x = ] using identities and/or algebra. Look to use identities to write the equation in terms of a single trig function. Look for algebraic manipulations such as factoring, combining fractions, etc. 2. After solving for a trigonometric function, solve for the variable over one period. 3. Write the general solutions, if necessary. [ie. ±k or ±2k] Note: NEVER Divide by a variable [x or ] or a function [sin x, or cos x, or tan x, …] You can use standard algebraic techniques to solve a trigonometric equation. Your goal is to isolate the trigonometric function involved in the equation, then solve for the angle.  0  x  2



Example #2: Solve the following equations for all values (in general) and for fundamental values  .  0    360 2 cos   1

a) sin x  cos x

b)

c) 4cos 2 x  1

d) sin x  2   sin x

e) 2 tan   3  0

f) csc2 x  10

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Chapter 5… Inverse Trigonometric Functions…Section 5.3

Trigonometry

Factoring with Trigonometric Functions Recall that the Zero-Product Property states that if the product of two numbers is zero, then at least one of the factors is equal to zero. Consider the following equations: Solve x2  x  20  0 Solve x2  3x  0 Solve 2x3  x2

If a trinomial cannot factor within an equation, then use the quadratic equation: x 

b  b 2  4 ac . 2a

Solve the following equation: 2 x2  4 x  1  0

Example #3: Find the exact solutions of the following equations in the interval [0, 2) as well as for all real values of x. a) 2cos 2 x  cos x  0 b) 2sin 2 x  3sin x  1  0

d) tan 2 x  1  0

c) sec x csc x  2 csc x

2

Chapter 5… Inverse Trigonometric Functions…Section 5.3

Trigonometry

Using Trigonometric Identities Look for Pythagorean Identities that can be substituted so that the equation is in terms of a single trig function. Example #4: Find all the solutions of the equation in the interval [0, 2 ) as well as for all real values of x. a) 3cos2 x  7sin x  7

b) csc2 x  4cot x  2

c) cos x  2 sec x  1

d) 2sec 2 x  tan x  3

Solving Trigonometric Equations with Sum/Difference and Double/Half Angle Identities Example #5: Find all the solutions of the equation in the interval [0, 2 ) as well as all real values of x. [Hint: Use a trigonometric identity first before solving.] a) cos 2 x  cos x  1  0 b) sin 2x + cos x = 0

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Chapter 5… Inverse Trigonometric Functions…Section 5.3

c) sin

Trigonometry

x  1  cos x 2

1 2

d) sin 2 x  sin 2 x

  1   e) sin  x    sin  x    6 6 2  

Functions Involving Multiple Angles Example #6: Find all the solutions of the trigonometric equation in the interval [0, 2 ) algebraically. a) 2cos3  1

  b) sin  2 x    1 3 

c) tan 3x (tan x  1)  0

d) 2 cos x sin 2 x  cos x

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