In general, if an angle whose measure is greater than has a reference angle of , , or , or if it is a quadrantal angle, we can find its ordered pair, and so we can find the values of any of the trig functions of the angle. The first step is to determine the reference angle.
Example 5: Find the value of each expression.
a.
b.
c.
Solution:
a.
is a full rotation of , plus an additional . Therefore the angle is coterminal with , and so it shares the same ordered pair, . The sine value is the coordinate.
b.
is two full rotations, or , plus an additional :
Therefore is coterminal with , so the ordered pair is . The tangent value can be found by the following:
c.
is a full rotation of , plus an additional . Therefore the angle is coterminal with , and the ordered pair is . So the cosine value is .
So far all of the angles we have worked with are multiples of and . Next we will find approximate values of the trig functions of other angles.
Trigonometric Function Values in Tables
As you work through this chapter, you will learn about different applications of the trig functions. In many cases, you will need to find the value of a function of an angle that is not necessarily one of the “special” angles we have worked with so far. Traditionally, textbooks have provided students with tables that contain values of the trig functions. Below is a table that provides approximate values of the sine, cosine, and tangent values of several angles.
Angle Cosine Sine Tangent
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We can use the table to identify approximate values.
Example 6: Find the approximate value of each expression, using the table above.
a.
b.
c.
Solution:
a.
We can identify the sine value by finding the row in the table for . The sine value is found in the third row of the table. Note that this is an approximate value. We can evaluate the reasonableness of this value by thinking about an angle that is close to , . We know that the ordered pair for is , so the sine value is , which is also in the table. It is reasonable that which is slightly less than the sine value of , given where the terminal sides of these angles intersect the unit circle.
b.
We can identify this cosine value by finding the row for . The cosine value is found in the second column. Again, we can determine if this value is reasonable by considering a nearby angle. is between and , and its cosine value is between the cosine values of these two angles.
c.
We can identify this tangent value by finding the row for , and reading the final column of the table. In the review questions, you will be asked to explain why the tangent value seems reasonable.
Using a Calculator to Find Values
If you have a scientific calculator, you can determine the value of any trig function for any angle. Here we will focus on using a TI graphing calculator to find values.
First, your calculator needs to be in the correct “mode.” In chapter 2 you will learn about a different system for measuring angles, known as radian measure. In this chapter, we are measuring angles in degrees. (This is analogous to measuring distance in miles or in kilometers. It’s just a different system of measurement.) We need to make sure that the calculator is working in degrees. To do this, press [MODE]. You will see that the third row says Radian Degree. If Degree is highlighted, you are in the correct mode. If Radian is highlighted, scroll down to this row, scroll over to Degree, and press [ENTER]. This will highlight Degree. Then press [Mode] to return to the main screen.
Now you can calculate any value. For example, we can verify the values from the table above. To find , press [SIN][130][ENTER]. The calculator should return the value .
You may have noticed that the calculator provides a “(” after the SIN. In the previous calculation, you can actually leave off the “)”. However, in more complicated calculations, leaving off the closing “)” can create problems. It is a good idea to get in the habit of closing parentheses.
You can also use a calculator to find values of more complicated expressions.
Example 7: Use a calculator to find an approximate value of . Round your answer to places.
Solution:
To use a TI graphing calculator, press [SIN][25][+][COS][25][ENTER]. The calculator should return the number . This rounds to
Lesson Summary
In this lesson we have examined the idea that we can find an exact or an approximate value of each of the six trig functions for any angle. We began by defining the idea of a reference angle, which is useful for finding the ordered pair for certain angles in the unit circle. We have found exact values of the trig functions for “special” angles, including negative angles, and angles whose measures are greater than . We have also found approximations of values for other angles, using a table, and using a calculator. In the coming lessons, we will use the ideas from this lesson to (1) examine relationships among the trig functions and (2) apply trig functions to real situations.
Points to Consider
What is the difference between the measure of an angle, and its reference angle? In what cases are these measures the same value?
Which angles have the same cosine value, or the same sine value? Which angles have opposite cosine and sine values?
Review Questions
State the reference angle for each angle.
State the ordered pair for each angle.
Find the value of each expression.
Find the value of each expression.
Find the value of each expression.
Use the table in the lesson to find an approximate value of
Use the table in the lesson to approximate the measure of an angle whose sine value is .
In example 6c, we found that . Use your knowledge of a special angle to explain why this value is reasonable.
Use a calculator to find each value. Round to places.
Use the table below or a calculator to explore sum and product relationships among trig functions.
Consider the following functions:
Do you observe any patterns in these functions? Are there any equalities among the functions? Can you make a general conjecture about and for all values of , ?
What about and ?
Use a calculator or your knowledge of special angles to fill in the values in the table, then use the values to make a conjecture about the relationship between and . If you use a calculator, round all values to places.
Use your knowledge of trigonometry to conjecture the value of the function:
Graph it and confirm or revise your prediction. What did you have to change, if anything?
Review Answers
Between and .
This is reasonable because
Conjecture:
Conjecture:
.
Vocabulary
Coterminal angles
Two angles in standard position are coterminal if they share the same terminal side.
Reference angle
The reference angle of an angle in standard position is the measure of the angle between the terminal side and the closest portion of the axis.
Relating Trigonometric Functions
Learning objectives
A student will be able to:
State the reciprocal relationships between trig functions, and use these identities to find values of trig functions.
State quotient relationships between trig functions, and use quotient identities to find values of trig functions.
State the domain and range of each trig function.
State the sign of a trig function, given the quadrant in which an angle lies.
State the Pythagorean identities and use these identities to find values of trig functions.
Introduction
In previous lessons we defined and worked with the six tr
ig functions individually. In this lesson, we will consider relationships among the functions. In particular, we will develop several identities involving the trig functions. An identity is an equation that is true for all values of the variables, as long as the expressions or functions involved are defined. For example, is an identity. In this lesson we will develop several identities involving trig functions. Because of these identities, the same function can have very many different algebraic representations. These identities will allow us to relate the trig functions’ domains and ranges, and the identities will be useful in solving problems in later chapters.
Reciprocal identities
The first set of identities we will establish are the reciprocal identities. A reciprocal of a fraction is the fraction . That is, we find the reciprocal of a fraction by interchanging the numerator and the denominator, or flipping the fraction. The six trig functions can be grouped in pairs as reciprocals.
First, consider the definition of the sine function for angles of rotation: . Now consider the cosecant function: . In the unit circle, these values are and . These two functions, by definition, are reciprocals. Therefore the sine value of an angle is always the reciprocal of the cosecant value, and vice versa. For example, if , then .
Analogously, the cosine function and the secant function are reciprocals, and the tangent and cotangent function are reciprocals:
We can use these reciprocal relationships to find values of trig functions. The fundamental identity stemming from the Pythagorean Theorem can take a great many new forms.
Example 1: Find the value of each expression using a reciprocal identity.
a.
b.
Solution:
a.
These functions are reciprocals, so if , then . It is easier to find the reciprocal if we express the values as fractions: .
b.
These functions are reciprocals, and the reciprocal of is .
We can also use the reciprocal relationships to determine the domain and range of functions.
Domain, Range, and Signs of Functions
While the trigonometric functions may seem quite different from other functions you have worked with, they are in fact just like any other function. We can think of a trig function in terms of “input” and “output.” The input is always an angle. The output is a ratio of sides of a triangle. If you think about the trig functions in this way, you can define the domain and range of each function.
Let’s first consider the sine and cosine functions. The input of each of these functions is always an angle, and as you learned in the previous chapter, these angles can take on any real number value. Therefore the sine and cosine function have the same domain, the set of all real numbers, . We can determine the range of the functions if we think about the fact that the sine of an angle is the coordinate of the point where the terminal side of the angle intersects the unit circle. The cosine is the coordinate of that point. Now recall that in the unit circle, we defined the trig functions in terms of a triangle with hypotenuse .
In this right triangle, and are the lengths of the legs of the triangle, which must have lengths less than , the length of the hypotenuse. Therefore the ranges of the sine and cosine function do not include values greater than one. The ranges do, however, contain negative values. Any angle whose terminal side is in the third or fourth quadrant will have a negative coordinate, and any angle whose terminal side is in the second or third quadrant will have a negative coordinate.
In either case, the minimum value is . For example, and . Therefore the sine and cosine function both have range from to .
The table below summarizes the domains and ranges of these functions:
Domain Range
Sine
Cosine
Knowing the domain and range of the cosine and sine function can help us determine the domain and range of the secant and cosecant function. First consider the sine and cosecant functions, which as we showed above, are reciprocals. The cosecant function will be defined as long as the sine value is not . Therefore the domain of the cosecant function excludes all angles with sine value , which are , etc.
In Chapter 2 you will analyze the graphs of these functions, which will help you see why the reciprocal relationship results in a particular range for the cosecant function. Here we will state this range, and in the review questions you will explore values of the sine and cosecant function or order to begin to verify this range, as well as the domain and range of the secant function.
Domain Range
Cosecant or
Secant or
Now let’s consider the tangent and cotangent functions. The tangent function is defined as . Therefore the domain of this function excludes angles for which the ordered pair has an coordinate of : , , etc. The cotangent function is defined as , so this function’s domain will exclude angles for which the ordered pair has a coordinate of : , , , etc. As you will learn in chapter 3 when you study the graphs of these functions, there are no restrictions on the ranges.
Function Domain Range
Tangent
Cotangent
Knowing the ranges of these functions tells you the values you should expect when you determine the value of a trig function of an angle. However, for many problems you will need to identify the sign of the function of an angle: Is it positive or negative?
In determining the ranges of the sine and cosine functions above, we began to categorize the signs of these functions in terms of the quadrants in which angles lie. The figure below summarizes the signs for angles in all quadrants.
Example 2: State the sign of each expression.
a.
b.
c.
Solution:
a. The angle is in the second quadrant. Therefore the coordinate is negative and so is negative.
b. The angle is in the third quadrant. Therefore the coordinate is negative. So the sine, and the cosecant are negative.
c. The angle is in the first quadrant. Therefore the tangent value is positive.
So far we have considered relationships between pairs of functions: the six trig functions can be grouped in pairs as reciprocals. Now we will consider relationships among three trig functions.
Quotient Identities
The definitions of the trig functions led us to the reciprocal identities above. They also lead us to another set of identities, the quotient identities.
Consider first the sine, cosine, and tangent functions. For angles of rotation (not necessarily in the unit circle) these functions are defined as follows:
Given these definitions, we can show that , as long as :
The equation is therefore an identity that we can use to find the value of the tangent function, given the value of the sine and cosine.
Example 3: If and , what is the value of ?
Solution:
Example 4: Show that
Solution:
This is also an identity that you can use to find the value of the cotangent function, given values of sine and cosine. Both of the quotient identities will also be useful in chapter 3, in which you will prove other identities.
Pythagorean Identities
The final set of identities that we will examine in this lesson are called the Pythagorean identities because they rely on the Pythagorean Theorem. In previous lessons we used the Pythagorean theorem to find the sides of right triangles. Consider once again the way that we defined the trig functions in lesson 4. Let’s look at the unit circle:
The legs of the right triangle are , and . The hypotenuse is . Therefore the following equation is true for all and on the unit circle:
Now remember that on the unit circle, and . Therefore the following equation is an identity:
Note: Writing the exponent after the cos and sin is the standard way of writing exponents. Just keeping mind that means and means .
We can use this identity to find the value of the sine function, given the value of the cosine, and vice versa. We can also us
e it to find other identities.
Example 5: If what is the value of ? Assume that is an angle in the first quadrant.
Solution:
Remember that it was given that is an angle in the first quadrant. Therefore the sine value is positive, so .
Example 6: Use the identity to show that
Solution:
Lesson Summary
In this lesson we have examined relationships between and among the trig functions. The reciprocal identities tell us the relationship between pairs of trig functions that are reciprocals of each other. The quotient identities tell us relationships among functions in threes: the tangent function is the quotient of the sine and cosine functions, and the cotangent function is the reciprocal of this quotient. The Pythagorean identities, which rely on the Pythagorean theorem, also tell us relationships among functions in threes. Each identity can be used to find values of trig functions, and as well as to prove other identities, which will be a focus of chapter 3. We can also use identities to determine the domain and range of functions, which will be useful in chapter 2, where we will graph the six trig functions.
CK-12 Trigonometry Page 5
