1. On a cold winter day the sun streams through your living room window and causes a warm, toasty atmosphere. This is due to the angle of inclination of the sun which directly affects the heating and the cooling of buildings. Noon is when the sun is at its maximum height in the sky and at this time, the angle is greater in the summer than in the winter. Because of this, buildings are constructed such that the overhang of the roof can act as an awning to shade the windows for cooling in the summer and yet allow the sun’s rays to provide heat in the winter. In addition to the construction of the building, the angle of inclination of the sun varies according to the latitude of the building’s location.
If the latitude of the location is known, then the following formula can be used to calculate the angle of inclination of the sun on any given date of the year:
where represents the number of the day of the year that corresponds to the date of the year.
a. Determine the measurement of the sun’s angle of inclination for a building located at a latitude of , March , the day of the year.
Note: This formula is accurate to
b. Determine the measurement of the sun’s angle of inclination for a building located at a latitude of , September , the day of the year.
2. A tower, high, must be secured with a guy wire anchored from the base of the tower. What angle will the guy wire make with the ground?
3. Using technology, graph and
This is the graph on the one-to-one function .
This is the graph of the inverse of the one-to-one function .
All of these functions can be graphed using the TI-83 graphing calculator. However, when doing the arcsecant, arccosecant and arccotangent functions, the and symbols are found under the [TEST] menu Math. As well the words and/or are in the same location under the [LOGIC] section of the [TEST] menu.
Lesson 2
Ranges of Inverse Circular Functions
Learning Objectives
A student will be able to:
Understand the ranges of the six circular functions and of their inverses.
Introduction
The graph of the equation is a circle with its center at the origin and a radius of one unit. Trigonometric functions are defined such that their domains are sets of angles and their ranges are sets of real numbers. Circular functions are defined such that their domains are sets of numbers that correspond to the measure of angles in radian units. Radian measure is the distance traveled on the unit circle after rotating about the circle for a given angle. So for a non-unit circle, it is the ratio of the arc length to the radius of the circle. where s is the length of the arc of the circle and r is the radius of the circle. All points on the unit circle have coordinates such that these coordinates are defined as the cosine and sine of the arc length from the intercept of to the point on the circumference of the unit circle. The arc length can be created by moving counter clockwise (positive) or clockwise (negative) from the intercept. Therefore, the domain of all of the circular functions is the set of real numbers. However, the ranges are more restricted. The remaining functions of tangent, cotangent, secant and cosecant can all be expressed in terms of sine and cosine by using the identities.
Domain and Range of the Circular Functions and their Inverses
The ranges of these circular functions, like their corresponding trigonometric functions, are sets of real numbers. These functions are called circular functions because radian measures of angles are determined by the lengths of arcs of circles. Trigonometric functions defined using the unit circle lead directly to these circular functions.
The graph of the equation is a circle in the rectangular coordinate system. This graph is called the unit circle and has its center at the origin and has a radius of . Trigonometric functions are defined so that their domains are sets of angles and their ranges are sets of real numbers. Circular functions are defined such that their domains are sets of numbers that correspond to the measures (in radian units) of the angles of trigonometric functions.
The following diagram begins with the unit circle . Point is located at the intersection of the unit circle and the axis. Let be any real number. Start at point and measure along the unit circle in a counterclockwise direction if and in a clockwise direction if , ending up at point . The sine and cosine of define the coordinates of point .
Sin and exist for each real number because are the coordinates of point located on the unit circle, that corresponds to an arc length of . Because this arc length can be positive (counterclockwise) or negative (clockwise), the domain of each of these circular functions is the set of real numbers. The range is more restricted. The cosine and sine are the coordinates of a point that moves around the unit circle, and they vary between negative one and positive one. Therefore, the range of each of these functions is a set of real numbers.
The domain and the range of these six circular functions and their inverses can be best understood by graphing the functions. The TI-83 will be used to graph all of the circular functions and their inverses.
The Graph of
For the circular function , the domain is the set of Real numbers and the range is
The Graph of
For the circular function , the domain is the set of Real numbers and the range is .
The Graph of
For the circular function , the domain is all Real numbers except and the range is the set of Real numbers. The tangent function can also be expressed as the quotient of , and when the value of equals zero, the tangent function is undefined (as is division by zero). As a result, the graph approaches infinity as it approaches these points.
The Graph of
For the circular function , the domain is all Real numbers except and the range is the set of Real numbers. since cotangent is the reciprocal of tangent, the graph approaches infinity when the value of equals zero.
The Graph of
For the circular function , the domain is all Real numbers except and the range is . The secant function has as its reciprocal function and the graph will approach infinity as it nears the points where equals zero.
The Graph of
For the circular function , the domain is all Real numbers except and the range is . The cosecant function has as its reciprocal function and the graph will approach infinity as it nears the points where equals zero.
Now we will examine the graphs of the inverse circular functions
The Graph of
For the inverse circular function, , the domain is and the range is
The Graph of
For the inverse circular function, , the domain is and the range is
The Graph of
For the inverse circular function, , the domain is the set of Real numbers and the range is
The Graph of
For the inverse circular function, , the domain is the set of Real numbers and the range is
The Graph of
For the inverse circular function, , the domain is and the range is
The Graph of
For the inverse circular function, , the domain is and the range is .
Now that all of the graphs have been created, the domains and ranges of the circular functions and their inverses should be evident. These values will be important when it comes to determining values for these functions.
Lesson Summary
You have seen the graphs of the circular functions and of their inverses as they are created using technology (TI-83). The notation used for indicating the domain and ranges of some of the functions is probably new. However, as you apply the values to the graphs, this new notation should become easier to remember.
Points to Consider
How do the values of the ranges of the inverse circular functions apply when values for these functions are determined?
Exact Values of Special Inverse Circular Functions
Learning Objectives
A student will be able to:
Use the point unit circle to determine exact values of special inverse circ
ular functions.
Introduction
In earlier lessons you learned about the reference triangles used to evaluate trigonometric functions for all integer multiples of and or and respectively. These values can be displayed on the unit circle or on the two special triangles.
The Unit Circle
The Special Triangles
Whichever format you prefer to become familiar with, the important thing is that you are able to use these reference diagrams to evaluate these special trigonometric functions.
Example 1: Use the special triangles or the unit circle to evaluate each of the following:
a.
b.
c.
d.
Solution:
a.
b.
c.
d.
Example 2: Use the special triangles or the unit circle to find the exact values of each of the following:
a.
b.
c.
Solution: Use the unit circle. Remember that the inverse sine and inverse tangent functions have values in , and the inverse cosine function has values in . Also, a point on the unit circle
a.
b.
c.
Lesson Summary
In this lesson, you learned how to use the unit circle to determine exact values of the inverse circular functions. When the unit circle is used to determine these values, the results are readily available in both degree and radian measure. These values are exact as compared to those obtained by using a calculator.
Points to Consider
Is it possible to apply the inverse composition rule to trigonometric functions?
Review Questions
Use the special triangles or the unit circle to evaluate each of the following:
Use the special triangles or the unit circle to find the exact values of each of the following:
Review Answers
Vocabulary
Unit Circle
A circle with its center at the origin and a radius of .
Recognize f(f-1(x)) = x and f-1(f(x)) = x (Range of the outside function, domain of the inside function)
Recognize and (Range of the outside function, domain of the inside function)
Learning objectives
A student will be able to:
Determine whether or not two functions are inverses by composing a function and its inverse.
Graph functions and . If the graphs are symmetric about the line , then the functions are inverses.
Introduction
Due to an unusually regular growth pattern, the population of a known region in Africa is given by the formula where is the population in thousands and is the number of years since 1970. What are the results of evaluating and ? What do these values mean with respect to the problem? This problem will help you to understand the definition of an inverse function and we will revisit it later in the lesson.
And
The statement means that . This relationship is used to determine values of . Suppose that is a function with the property that each value of determines one and only one value of . Then has an inverse function, and if and only if Let’s take a closer look at this general definition of an inverse function by graphing a function and its inverse.
Given, create a table of values.
To create a table of values for , the columns of the first table can be simply interchanged.
Both tables contain the same values, but with the columns interchanged. Therefore, the relationship between the function and can be demonstrated by exploring a few of the values.
and hence
and hence
The above result will hold true for any input , so for all values of for which is defined.
Likewise, and
The above result will hold true for any , so for all values of for which is defined.
Using technology, the graphs of the function and its inverse can be created without finding a formula for the inverse. Technology will graph the inverse by entering a command – Draw Inverse.
The red curve represents the function . The dotted line is the graph of . The curve below is the mirror image of the graph of . If the axis and the axis have the same scale, the point is reflected on the mirror image as It is evident that these graphs are symmetrical across the line . The function is an exponential function that has all the real numbers as its domain and all the positive numbers as its range. The function has all the positive numbers as its domain and all the real number as its range. In other words, the domain of is equal to the range of and the domain of is equal to the range of .
Now, let’s revisit the problem at the beginning of the lesson.
To evaluate :
This means that in the year 2005 the population was people.
To evaluate
so in , represents the population and is the year in which the population was people.
This means that it took 25 years for the population to reach people and this occurred in the year 1995.
Now that you have a better understanding of a function and its inverse, we will apply this knowledge to some questions to prove whether or not two given functions are indeed inverses and to determine the inverse of a given function.
Example 1: Prove that and are inverse functions and show the results graphically on the same axes with the same scale.
Solution:
Graphing:
The red graph is the graph of and the black graph is the graph of . Both graphs are symmetric about the dotted line .
Example 2: What is the inverse of the function?
Solution:
Let and solve for .
If this function is the inverse of then their graphs should be mirror-images about the line .
Graphing the function and its inverse is a way to check the solution. The red graphs is and the blue one is
Leson Summary
In this lesson you learned a very important property about functions and their inverses. This property included the statements for all values of for which is defined and for all values of for which is defined. You have also learned how to determine an inverse of a given function algebraically, how to prove algebraically that functions are invertible and how to prove graphically that functions are inverses.
Points to Consider
Can this property be applied to derive properties about other functions and their inverses?
Review Questions
Use a graph to determine whether or not the following functions are invertible. Explain the results of each graph.
Prove that the following functions are inverses.
and
Review Answers
(a) The function is not invertible because the inverse is not a mirror-image about the line .
(b)
The function is invertible because the inverse is a mirror-image about the line .
Prove that and algebraically. Following is the correct way to begin the proof:
Vocabulary
Inverse Function
Two functions are inverse functions if and only if for all values of .
Invertible
If a function has an inverse, it is invertible.
Applications, Technological Tools
Learning objectives
A student will be able to:
Use technology to graph functions and their inverses.
Solve world problems using the fact that .
Introduction
The following problem that involves functions and their inverses will be solved using the property . In addition, technology will also be used to complete the solution.
1. To commemorate the centennial of the flight of the Silver Dart, an exact replica was built and was successfully flown on Baddeck Bay on Sunday, February 22, 2009. One of the attempts saw the plane fly successfully feet before it lost a wheel and landed on the frozen Bay. The following parabola is a graph of the plane’s height, , in feet as a function of time, , in minutes.
a. A
pproximately, what is the maximum height reached by the plane?
b. Approximately, when did the Silver Dart land on Baddeck Bay?
c. Restrict the domain of so that has an inverse. Graph this new function with the restricted domain.
d. Graph the inverse of the function from part (c).
e. Rewrite the problem to reflect the new function from part (c).
Solution:
a. The maximum height reached by the plane is approximately .
b. The Silver Dart landed on Baddeck Bay after becoming airborne.
c. The axis of symmetry for the parabola that depicts the flight is . Therefore the domain of the right half of the parabola is the interval .
CK-12 Trigonometry Page 15