CK-12 Trigonometry
Page 14
So, either or ,
Move over to the other side and use the sum-to-product formula:
So, either
Using the sum-to-product formula:
Derive a formula for .
Summary and Review of Trigonometric Identities
The sum and difference identities, the double and half angle identities, and the product to sum and sum to product identities derived and discussed in this section help to expand the set of angles for which trigonometric functions can be obtained. For example knowing the values for the sine and cosine of and , and the sum and difference formulas for sine and cosines gives us the opportunity to know the fractional values for sine and cosine of , that is or , that is or the , that is . Using the key angles for the first quadrant , what are some of the other classes if angles can be found? Then list some of the angles that can be obtained using key angles from other quadrants. Here are the identities studied in this chapter to help with this culminating exercise.
Chapter Review Exercises
Find the sine, cosine, and tangent of an angle with terminal side on .
If and , find .
Simplify: .
Verify the identity:
Find all the solutions in the interval .
Find the exact value of:
Write as a product:
Simplify:
Simplify:
Derive a formula for
Answers
If the terminal side is on , then the hypotenuse of this triangle would be (by the Pythagorean Theorem, ). Therefore, , , and .
If and tan then is in Quadrant II. Therefore is negative. To find the third side, we need to do the Pythagorean Theorem.
Factor top, cancel like terms, and use the Pythagorean Theorem Identity.
Change secant and cosecant into terms of sine and cosine, then find a common denominator.
So, or
or
Use the half angle formula with .
Use the sine sum formula.
Use the sine and cosine sum formulas.
Use the sine sum formula as well as the double angle formula.
Chapter 4: Inverse Functions and Trigonometric Equations
Lesson 1
General Definitions of Inverse Trigonometric Functions
Learning Objectives
A student will be able to:
Relate the knowledge of inverse functions to trigonometric functions.
Understand and evaluate inverse trigonometric functions.
Introduction
A new outdoor skating rink has just been installed outside a local community center. A light is mounted on a pole above the ground. The light must be placed at an angle so that it will illuminate the end of the skating rink. If the end of the rink is from the pole, at what angle of depression should the light be installed? This problem differs from other trigonometry problems we have seen so far. The standard trigonometric functions take angles as inputs, and give ratios between the sides of a triangle. In this problem we are given information about the sides of a triangle and need to use them to solve for the angle. This means we need a function which does the opposite – or inverse.
Inverse Functions
In a previous lesson, you learned that each function has an inverse relation and that this inverse relation is a function only if the original function is one-to-one. A function whose inverse is a function will have a graph that passes both the vertical line test and the horizontal line test. Each line will intersect the graph in one place only.
This is the graph of The graph suggests that is one-to-one. It passes both the vertical and the horizontal line tests. If is one-to-one, the inverse function will satisfy the equation .
This can be proven algebraically. (Switch the and the )
Therefore “ inverse” or
The symbol is read “ inverse” and should not be read as the reciprocal of “”. The reciprocal of must be written as . Determining an inverse function algebraically can be both involved and difficult. Therefore, determining an inverse function will be done by applying what we know about mapping to and mapping to . The graph of can be used to produce the graph of by applying the inverse reflection principle:
The points and in the coordinate plane are symmetric with respect to the line .
The points and are reflections of each other across the line .
This is the graph of the . It is also a one-to-one function.
If you study the graph in figure 1, it is obvious that the inverse reflection principle is shown here.
Not all functions have inverses that are one-to-one. However, the inverse can be modified to a one-to-one function if a “restricted domain” is applied to the inverse function. This concept of “restricted domains” will be vital when we examine the inverse functions of the trigonometric functions.
Inverse Trigonometric Functions
It is time to return to the situation that was presented in the introduction. The first step is to draw a proper diagram to represent the problem.
In this diagram, the angle of depression which is located outside of the triangle, is not known. However, the angle of depression equals the angle of elevation. (Remember from Geometry, when two parallel lines are crossed by a transversal, opposite internal angles are congruent.)
For the angle of elevation, the pole where the light is located is the opposite and is high. The length of the rink is the adjacent side and is in length. To calculate the measure of the angle of elevation the trigonometric ratio for tangent can be applied.
The angle of depression at which the light must be placed to light the rink is
The trigonometric value tan of the angle is known, but not the angle. In this case the inverse of the trigonometric function must be used to determine the measure of the angle. This function is located above the [tan] button of the calculator. To access this function, press tan and the measure of the angle appears on the screen Notice the notation . This inverse of the tangent function is the arctan relation. The inverse of the cosine function is the arccosine relation (also called the arccos relation) and the inverse of the sine function is the arcsine relation (also called the arcsin relation). Let’s consider another example:
Example 1:
A deck measuring by will require laying boards with one board running along the diagonal and the remaining boards running parallel to that board. The boards meeting the side of the house must be cut prior to being nailed down. At what angle should the boards be cut?
Solution:
The boards should be cut at an angle of .
Example 2:
You live on a farm and your chore is to move hay from the loft of the barn down to the stalls for the horses. The hay is very heavy and to move it manually down a ladder would take too much time and effort. You decide to devise a make shift conveyor belt made of bed sheets that you will attach to the door of the loft and anchor securely in the ground. If the door of the loft is above the ground and you have of sheeting, at what angle do you need to anchor the sheets to the ground?
Solution:
The sheets should be anchored at an angle of .
Lesson Summary
You have learned that each function has an inverse relation, and that the inverse relation is a function only if it is a one-to-one function. A one-to-one function passes both a vertical line test and a horizontal line test. A formula for the inverse of a function can be determined algebraically but the process can often be complex. Therefore, the knowledge that we know about functions and their inverse will be applied to determine the . The inverse of the trigonometric functions can be used to calculate the measure of an unknown angle in a triangle. The graphing calculator is an asset to performing this task.
Points to Consider
Are the inverse relations of the six basic trigonometric functions one-to-one?
Is there an interval on which these inverse relations are one-to one functions?
What are the restricted domains for the inverse relations of t
he trigonometric functions?
Review Questions
Study each of the following graphs and answer these questions: Is the graphed relation a function?
Does the relation have an inverse that is a function?
A foot ladder is leaning against a wall. If the foot of the ladder is from the base of the wall, what angle does the ladder make with the floor?
Review Answers
i) The graph represents a one-to-one function. It passes both a vertical and a horizontal line test. At this point we will have to say that the inverse of this relation is not a function.
ii) The graph represents a one-to-one function. It passes both the vertical and horizontal line tests.
It does not have an inverse that is a function.
iii) The graph does not represent a one-to-one function. It fails a vertical line test.
It does have an inverse that is a function.
Vocabulary
One-to-one function
a function that passes both the vertical line test and the horizontal line test.
arccosine
the inverse function of .
arcsine
the inverse of .
arctangent
The inverse of .
Using the “inverse” notation
Using the “inverse” notation:
Learning Objectives
A student will be able to:
Understand the inverse sine function, inverse cosine function and the inverse tangent function.
Extend the inverse trigonometric functions to include the and functions.
Understand the meaning of restricted domain as it applies to the inverses of the six trigonometric functions.
Apply the domain, range and quadrants of the six inverse trigonometric functions to evaluate expressions.
Introduction
The function is not one-to-one and therefore has no inverse. In the following graph of , the graph fails the horizontal line test.
Inverse Trigonometric Equations
In order to consider the inverse function, we need to restrict the domain so that we have a section of the graph that is one-to-one. If the domain of is restricted to a new function . is defined. This new function is one-to-one and takes on all the values that the function takes on. Since the restricted domain is smaller, takes on all values once and only once.
In the previous lesson the inverse of was represented by the symbol , and The inverse of will be written as . or .
In this lesson we will use both and and both are read as “the inverse sine of ”or “the number between and whose sine is .”
The graph of is obtained by applying the inverse reflection principle and reflecting the graph of in the line . The domain of becomes the range of , and hence the range of becomes the domain of .
Another way to view these graphs is to construct them on separate grids. If the domain of is restricted to the interval , the result is a restricted one-to one function. The inverse sine function is the inverse of the restricted section of the sine function.
The domain of is and the range is .
The restriction of is a one-to-one function and it has an inverse that is shown below.
The statements and are equivalent for values in the restricted domain and values between and .
The domain of is and the range is .
The inverse functions for cosine and tangent are defined by following the same process as was applied for the inverse sine function. However, in order to create one-to-one functions, different intervals are used. The cosine function is restricted to the interval and the new function becomes . The inverse reflection principle is then applied to this graph as it is reflected in the line The result is the graph of (also expressed as ).
Another way to view these graphs is to construct them on separate grids. If the domain of is restricted to the interval , the result is a restricted one-to one function. The inverse cosine function is the inverse of the restricted section of the cosine function.
The domain of is and the range is .
The restriction of is a one-to-one function and it has an inverse that is shown below.
The statements and are equivalent for values in the restricted domain and values between and .
The domain of is and the range is .
The tangent function is restricted to the interval and the new function becomes . The inverse reflection principle is then applied to this graph as it is reflected in the line . The result is the graph of (also expressed as .
Another way to view these graphs is to construct them on separate grids. If the domain of is restricted to the interval , the result is a restricted one-to one function. The inverse tangent function is the inverse of the restricted section of the tangent function.
The domain of is and the range is .
The restriction of is a one-to-one function and it has an inverse that is shown below.
The statements and are equivalent for values in the restricted domain and values between and .
The domain of is and the range is .
The above information can be readily used to evaluate inverse trigonometric functions without the use of a calculator. These calculations are done by applying the restricted domain functions to the unit circle.
Example 1:
Find the exact value of each expression without a calculator.
a.
b.
c.
Solution:
a. Sketch a diagram that shows the point on the unit circle (right half) that has as its coordinate. Draw a reference triangle.
From the diagram, you can see that this is one of the special ratios.
The angle in the interval whose sine is is .
In other words, .
b. Follow the same steps as in the solution of part a. The point on the unit circle (top half) will have as its coordinate.
From the diagram, you can see that this is one of the special ratios.
The angle in the interval whose cosine is is .
In other words, .
c. Follow the same steps as in the solution of part a. The point on the unit circle (right side) will have times its coordinate as its coordinate.
From the diagram, you can see that this is one of the special ratios.
The angle in the interval whose tangent is is .
In other words, .
Inverse Trigonometric Functions
Restricted Domain Function Inverse Trigonometric Function Domain Range Quadrants
AND
AND
AND
AND
AND
AND
AND
AND
AND
AND
All Real Numbers
AND
AND
Now that the six trigonometric functions and their inverses have been summarized, let’s take a look at the graphs of the six inverse trigonometric functions.
The above graphs of the inverse trigonometric functions are from the website: www.intmath.com/Analytic-trigonometry/7
Lesson Summary
In this lesson you have learned that although the trigonometric functions, as you them, are not one-to-one, it is very important to study their inverses. The trigonometric functions can be made one-to-one by simply restricting the domain of the original function to one that creates the one-to-oneness. The graphs of each of these restricted domain functions are readily created using the graphing calculator or by using suitable software. The results were then applied to evaluating inverse function values without the use of a calculator.
Points to Consider
Can the values of the special angles of the unit circle be applied to the inverse trigonometric functions?
Is it possible to determine exact values for the special inverse circular functions?
Review Questions
Determine the exact value of the following expressions without using a calculator. Provide a sketch to illustrate each expression.
&n
bsp; Review Answers
Does not exist. is greater than and the domain of is
Applications, Technological Tools
Learning Objectives
A student will be able to:
Use technology to graph the inverse trigonometric functions.
Solve real world problems using the inverse trigonometric functions.
Introduction
The following problems are real-world problems that can be solved using the trigonometric functions. In everyday life, indirect measurement is used to obtain answers to problems that are impossible to solve using measurement tools. However, mathematics will come to the rescue in the form of trigonometry to calculate these unknown measurements. In addition to solving problems, we will also use the graphing calculator to produce graphs of these functions.