CK-12 Trigonometry
Page 16
d.
e. Answers will vary. One response could reflect that the pilot preformed a stunt at into the flight at a height of and then immediately descended for a landing at .
The TI-83 was used to create the graphs.
Example 1: Find the inverse of the following trigonometric functions:
a.
b.
Solution:
a.
If these are inverses, then the graphs should be reflections about the line . The following graph shows that this is true and that the inverse of the function is
Example 2:
Find the inverse of the trigonometric function
Solution:
The following graph shows that the inverse of the trigonometric function is indeed
Lesson 3
Derive Properties of Other Five Inverse Circular Functions in terms of Arctan(short)
Learning Objectives
A student will be able to:
Relate the concept of inverse functions to trigonometric functions.
Compose each of the six basic trigonometric functions with .
Reduce the composite function to an algebraic expression involving no trigonometric functions.
Introduction
In the previous lesson you learned that for a function for all values of for which is defined. If this property is applied to the trigonometric functions, the following equations will be true whenever they are defined:
a.
b.
c.
As well, you learned that for all values of for which is defined. If this property is applied to the trigonometric functions, the following equations that deal with finding an inverse trig. function of a trig. function, will only be true for value of within the restricted domains.
a.
b.
c.
These equations are better known as composite functions and are composed of one trigonometric function in conjunction with another different trigonometric function. The composite functions will become algebraic functions and will not display any trigonometry. Let’s investigate this phenomenon.
Composing Trigonometric Functions with Arctan
Let’s express trig functions in a new way – one that works better with arctan than with any of the other inverse functions. To begin this investigation, we will draw a representation of the tangent function as it appears on the unit circle. The unit circle is the circle with its center at the origin and a radius of . Angle is formed by rotating about the origin to . Point is the intersection of line and the line .
The vertical line is a tangent to the unit circle (a tangent is a line that touches a curve at only one point). is the point where the diagonal line meets the tangent line and is where the tangent line meets the circle and the horizontal line. The line is called the tangent of or tan . The angle is the coordinate in the ordered pair . The value of the tangent is the slope of the line (the terminal side of angle in standard position).
Now that you understand the meaning of the tangent function and its importance to trigonometric functions, we will continue by first drawing a triangle that has measured in radians such that .
The hypotenuse of the right triangle can be determined by using the Pythagorean Theorem.
Using the triangle, all of the required ratios can be written as algebraic expressions with no trig. functions. To do this, remember . You will also have to recall:
If , then is a negative angle in quadrant IV. This same process can be used to compose trigonometric functions with the other basic inverse functions. However, the triangle that you begin with will have to be different because will have to equal or . You can explore these on your own and check your results with a classmate. Now, let’s apply the results of this investigation to some exercises.
Example 1: Without using technology, find the exact value of each of the following:
a.
b.
c.
Solution: Use the unit circle or special triangles to determine the exact values.
a.
b.
c.
Example 2: In the main concourse of the local arena, there are several viewing screens that are available to watch so that you do not miss any of the action on the ice. The bottom of one screen is above eye level and the screen itself is high. The angle of vision (inclination) is formed by looking at both the bottom and top of the screen.
a. Sketch a picture to represent this problem.
b. Calculate the measure of the angle of vision that results from looking at the bottom and then the top of the screen. At what distance from the screen does this value of the angle occur?
Solution:
a.
To determine these values, use a graphing calculator and the trace function to determine when the actual maximum occurs.
From the graph, it can be seen that the maximum occurs when . and .
Lesson Summary
In this lesson you learned how to find inverse trig functions of trig functions. The surprise was that these reduced to purely algebraic expressions- not ones that involved standard trigonometric functions
Points to Consider
Is it possible to graph these composite functions?
If so, is it possible to analyze the graphs.
Review Questions
Express each of the following functions as an algebraic expression involving no trigonometric functions.
Graph the function and state its domain and range.
Review Answers
The domain is all of the real numbers except where is an integer.
The range is .
Derive Inverse Cofunction Properties (short)
Learning objectives
A student will be able to:
Understand the cofunction identities.
Use the cofunction identities to prove identities.
Introduction
Recall that two angles are complementary if their sum is . In every triangle, the sum of the interior angles is and the right angle has a measure of . Therefore, the two remaining acute angles of the triangle have a sum equal to , and are complementary angles. Let’s explore this concept to identify the relationship between a function of one angle and the function of its complement in any right triangle. In other words, let’s explore the cofunction identities. A cofunction is a pair of trigonometric functions that are equal when the variable in one function is the complement in the other.
Cofunction Identities
In , is a right angle, and are complementary.
The Trigonometric Ratios with respect to are:
The Trigonometric Ratios with respect to are:
The value of a function with respect to is identical to the value of its cofunction with respect to . Therefore the following statements are true:
and for each of the above . The sine and cosine functions are cofunctions so:
and
The tangent and cotangent functions are cofunctions so:
and
The cosecant and secant functions are cofunctions so:
and
The following graph represents two complete cycles of the sinusoidal curve and of the sinusoidal curve .
Using the cofunction identities, the function , can be written as .
The graph of this function is shown below:
Notice that the two graphs are identical. The equation for the second graph that was entered as begins to the right of the graph of . In other words, the graph of is simply the graph of has undergone a phase shift or a horizontal translation of .
These cofunction identities hold true for all real numbers for which both sides of the equation are defined. Now that we have derived these new identities, it is time to see them in action.
Example 1:
1. Use the cofunction identities to evaluate each of the following expressions:
a. If determine
b. If determine .
Solution:
a. therefore
b. the
refore
Example 2:
2. Prove .
Solution:
Lesson Summary
In this lesson you learned the derivation of the cofunction identities. In the examples, you were able to see the application of the identities. These identities are used in trigonometry to prove identities and to derive other formulas that are used in solving trigonometric equations.
Points to Consider
Is there a relationship between the three basic trigonometric functions and their reciprocal functions.
Review Questions
Prove
If find
Review Answers
Vocabulary
Cofunction
a pair of trigonometric functions that are equal when the variable in one function is the complement in the other.
Complementary angles
two angles whose sum is .
Inverse Reciprocal Properties
Learning objectives
A student will be able to:
Understand the Inverse Reciprocal Properties of Inverse Trigonometric Functions.
Introduction
In previous lessons you learned the three basic trigonometric functions and their reciprocals. The reciprocal of is . and . The product of these reciprocals is one which is true for the definition of reciprocal. The other reciprocal functions are and and likewise and . Now, let’s apply the definition of reciprocal to the reciprocals of the inverse trigonometric function, keeping in mind the does not mean the reciprocal of but rather the inverse of the sine function.
Inverse Reciprocal Functions
We already know that the cosecant function is the reciprocal of the sine function. This will be used to derive the reciprocal of the inverse sine function.
The other inverse reciprocal identities can be proven by using the same process as above. However, remember that these inverse functions are defined by using restricted domains and the reciprocals of these inverses must be defined with the intervals of domain and range on which the definitions are valid. The remaining inverse reciprocal identities are:
For use on the calculator, the following conversion identities are used:
Now, let’s apply these identities to some problems that will give us an insight into how they work.
Example 1:
Evaluate
Solution:
Example 2:
2. Evaluate .
Solution:
is in the interval
Lesson Summary
In this lesson you learned the identities for the inverse reciprocal trigonometric functions. The most difficult part in applying these identities is remembering the domain and range that is applicable to each function. These identities are used to evaluate trigonometric expressions as shown above.
Points to Consider
Do exact values of functions of Inverse functions exist if Pythagorean triples are used?
Review Questions
Evaluate each of the following:
Review Answers
Find Exact Values of Functions of Inverse Functions using Pythagorean Triples. Repeat with non-integer values with calculator...
Learning objectives
A student will be able to:
Find exact values of functions of inverse functions using Pythagorean triples.
Introduction
A right triangle has sides and where and are the legs of the triangle and is the hypotenuse. This leads to the Pythagorean Theorem
If and are positive integers, this is called a Pythagorean triple. The smallest and best known example of a Pythagorean triple is
The integers and satisfy the Pythagorean Theorem.
Pythagorean Triples and Exact Values
A Pythagorean triple can consist of three even numbers or two odd numbers and one even number. It can never consist of three odd numbers or two even numbers and one odd number. The reason for this is the square of an odd number is odd and the square of an even number is even. The sum of two even numbers is an even number and the sum of an odd number and an even number is an odd number. Therefore, if either or is odd, then the other must be even and this would make an odd number. If both and are even numbers, then this would make an even number also. There are many ways to generate Pythagorean triples, but here is one method that will work all the time.
If and are two positive integers such that , then the values of and can be determined by using the following formulas:
Before we explore any further, let’s substitute the formulas for and into the Pythagorean Theorem to determine if they satisfy the theorem.
Both sides of the equation are equal which tells us that the formulas satisfy the Pythagorean Theorem. Therefore, the formulas can be used to generate Pythagorean triples.
Example 1:
If two positive integers and are given such that and generate a Pythagorean triple for these integers.
Solution:
The Pythagoren triple is .
Check the values of and using the Pythagorean Theorem.
Now that you are able to generate a Pythagorean triple, let’s determine the exact values of functions of inverse functions using a Pythagorean triple.
The above triangle represents the most common Pythagorean triple and we will use this to determine exact functions of inverse functions.
Example 2:
Evaluate .
Solution:
Let
and is in the Quadrant
Example 3:
Evaluate
Solution:
Let
and is in the Quadrant IV.
Now, let’s use our calculator and do the same questions.
Example 4:
Solution:
Using the TI-83 calculator
Example 5:
Solution:
Using the TI-83 calculator
Lesson Summary
In this lesson you learned how to generate Pythagorean triples by using simple formulas for and . You also used these values to evaluate functions of inverse functions by using both the right triangles and technology.
Points to Consider
Can these inverse circular functions be applied to other concepts that we have learned previously?
Review Questions
Evaluate the following using the Pythagorean triple
and it is in Quadrant I
Check using technology:
Lesson 4
Revisiting y = c + a cos b(x - d)
Learning objectives
A student will be able to:
Understand the graph of and its transformations.
Solve for in terms of to calculate values of given specific values for
Introduction
A remote control helicopter was being tested for its consistent flying ability.Under correct monitoring; the helicopter could fly up and down in a sinusoidal pattern. To demonstrate this movement, a graph was drawn to show the helicopter’s height at various times. The graph showed that the helicopter reached its maximum height of in and at it was at its minimum height of . We will revisit this scenario later in the lesson.
Consider the graph of
You will notice that the graph has an amplitude of , a period of , a sinusoidal axis of and no phase shift. However, all of these parts of the cosine curve can undergo transformations that will change the graph. The general form of the cosine curve that includes all of the transformations is . The letter represents the amplitude of the function. The amplitude is the and the is the vertical stretch of the graph. The letter represents the stretching or shrinking (horizontal stretch) of the graph along the axis. The following relationship exists between and the period of the graph: and solving for the period . The letter represents the phase shift (horizontal translation) of the graph along the axis. The phase shift of the graph will be to the left if is negative and if it is positive the shift will be to the right. The letter represents the vertical translation of the graph and will affect the location of the sinusoida
l axis.
Transformations of y = cos x
The following graphs will be used to see the affect that each of these transformations have on the graph of .
A vertical translation of will cause the graph to slide vertically upward if the value of is positive and downward if the value of is negative. This affects sinusoidal axis and its equation will no longer be . The following figure displays the graph of and the graph
The value of is and the graph of has moved upward such that the equation of the sinusoidal axis is now .
A vertical stretch of will cause the graph stretch vertically. This affects the amplitude of the graph and it will no longer be one. The following figure displays the graph of and the graph