7.13
Throughout your study of algebra, you have encountered algebraic statements that are always true,
sometimes true, or never true. Earlier in this unit, you learned about the trigonometric
functions y = sin(x) , y = cos(x) , and y = tan(x). You learned how to represent these functions in a
table, on a graph, and with a unit circle. In the next lessons, you will expand your understanding of
trigonometric functions and develop strategies to determine when trigonometric equations are true.
All answers in this unit should be given in radians unless otherwise stated in the problem.

12-2. WHEN IS IT TRUE?

Your teacher will assign your team one of the following equations. Your task is to decide if the
equation is always true, sometimes true, or never true, and to justify your decision using as many
representations as you can.

If the equation is always true, explain how you know. Then look for ways you could change
the equation so that it would never be true.

If the equation is sometimes true, find the exact values of x that make it true. Then decide
how you could change the equation so that it would never be true.

If the equation is never true, explain how you know. Then decide how you could change the
equation so that it would sometimes be true.
a. sin(x) =
b. cos (x) = 2
c. tan (x) = 0
d. sin (x) =
In the previous problem, you recognized that an equation could be always true, sometimes true, or
never true. In the next problems you will focus on trigonometric equations that are sometimes true,
as you learn how to solve them and how to determine the number of solutions they have.

12-23. When is the equation cos(x) =
true?
a. What do the solutions to this equation represent?
b. Solve the equation graphically. How many solutions can you find on the graph?
c. Draw a unit circle showing the solutions to cos(x) =

. How many solutions do you see?
12-24. Just as taking the square root (
) can be used to undo a square when solving an equation
algebraically, there is an undo (inverse) operation for cosine.
a. With your team, see if you can find the undo operation for cosine on your calculator. What
does it look like? How can you be sure that it is undoing cosine?
b. Use the undo operation for cosine on your calculator to solve the equation cos(x) = . Show
where the solution given by the calculator can be seen on the graph and on the unit circle.

12-25. With your team, decide how many solutions there are to the equation cos(x) = . Justify
your decision using as many representations as you can. How can you write all of them?

12-26. How do all of the solutions to cos(x) =
calculator?

12-27. Find all of the solutions for each equation below. You may use your calculator, but you must
represent your solutions graphically and on a unit circle. After you have found all of the solutions,
identify which ones lie in the domain 0 ≤ x ≤ 2π. Then work with your team to write a summary
statement about your solutions.
relate to the one given by cos−1 = ( ) on your
a. 2cos(x) + 1 = 0
b. tan(x) = 1

12-28. Jeremy used the sin−1 button on his calculator to solve a trigonometric equation and got the
solution 37°. He knows that there must be more than one solution. What are the rest of the solutions
to Jeremy’s equation? Explain how you found them. Use a unit circle and a graph in your
explanation.
You just learned how to use the inverse trigonometric functions to undo operations and solve
trigonometric equations. You also learned that the keys for the inverse trigonometric functions on
your calculator give you only one of the possible solutions to these equations. In these next
problems, you will investigate the inverse trigonometric functions more thoroughly. You will
determine whether the inverses are also functions.

12-38. Obtain the Lesson 12.1.3 Resource Page from your teacher.
a. Make a careful graph of y = sin(x). Is it a function? What are its domain and range?
b. Graph the line y = x.
c. Reflect the graph of y = sin(x) across the line y = x to make a graph of its inverse, x = sin(y).
Is it a function? What are its domain and range?
d. Discuss with your team the restriction you would need to make on the domain of y = sin(x)
so that its inverse is a function.

12-39. Now graph y = sin−1(x) on your graphing calculator. How is the result different from the
graph that you made? Does it make sense in terms of your answer to part (d) in the problem above?
Highlight the portion of your graph that represents the inverse function and label it with its domain
and range. Also highlight the portion of the unit circle (provided on the resource page) that shows
the restriction on the domain of y = sin(x) required to make its inverse a function.

12-40. Consider the equation
.
a. How many solutions does the equation have?
b. Without using a calculator, find the solutions and show them on the unit circle.
c. Which solution do you predict your calculator will give? How do you know? Once you have
made your prediction, use your calculator to solve
that this solution can be written as

. Were you correct? (Note
.)
12-41. Obtain two more copies of the Lesson 12.1.3 Resource Page from your teacher. Sketch
y = cos(x) and y = tan(x) on separate axes and label the domain and range of each.
a. Reflect each graph across the line y = x to generate graphs of the inverse relations.
b. Find the domain and range of the relations x = cos(y) and x = tan(y).
c. Use your graphing calculator to graph the functions y = cos−1(x) and y = tan−1(x). Highlight
the portion of your graph that represents the inverse function. Find the domain and range for
each function. Then highlight a portion of the unit circle (provided on the resource pages)
that shows the domain restrictions required to make the inverses of y = cos(x) and y = tan(x)
function.

12-42. For each of the following problems, find all of the solutions without using a calculator.
Draw a graph or unit circle to support your answers. Then predict the solution that your calculator
will give and use your calculator to check your prediction.
a.
c.
b.
d.