P.o.D. – Solve over the interval
[0,2𝜋]
1.) tan 𝑥 = 1
2.) cos 𝑥
√3
=−
2
1
3.) sin 𝑥 =
4.) cos 𝑥 =
𝜋 5𝜋
1.) ,
4
2.)
4
5𝜋 7𝜋
6
,
6
2
√3
2
𝜋 5𝜋
3.) ,
6
6
𝜋 11𝜋
4.) ,
6
6
4.7 – Inverse Trigonometric
Functions
Learning Target: I will be able to
evaluate and graph inverse trig
functions.
Arc Functions:
𝑦 = arcsin 𝑥 𝑜𝑟 𝑦 = 𝑠𝑖𝑛−1 𝑥
Domain: -1<x<1
𝜋
𝜋
2
2
Range: − ≤ 𝑦 ≤
Graph arcsine.
*Have your table of
Trigonometric Values {Your Unit
Circle Chart} readily available.
EX: Find the exact value of
arcsin(-1).
−𝜋 3𝜋
=
2
2
EX: Find the exact value of
−1 1
𝑠𝑖𝑛 ( ).
2
Q1:
𝜋
6
𝜋
5𝜋
6
6
Q2: 𝜋 − =
EX: Find the exact value of
𝑠𝑖𝑛−1 (√3).
No Solution. We can NOT take
the arcsine or arccosine of a
value greater than 1.
EX: A 12’ flagpole casts a shadow
of 22’. What is the angle of
elevation to the sun? (Write your
answer in degrees)
(Draw a triangle on the
whiteboard – show all work)
About 28.6 degrees
EX: Sketch a graph of y=arcsin(2x)
(draw a detailed graph on the
whiteboard)
EX: Find the exact value of
√3
arccos .
2
𝜋
Q1:
6
Q4:
11𝜋
6
EX: Find the exact value of
𝑐𝑜𝑠
−1
1
(− ).
2
Q2:
2𝜋
3
𝜋
4𝜋
3
3
Q3: 𝜋 + =
EX: Find the exact value of
arctan(1).
Q1:
𝜋
4
𝜋
5𝜋
4
4
Q3: 𝜋 + =
EX: Find the exact value of
−1 √3
𝑡𝑎𝑛 ( ).
3
*Remember, tangent is (+) in Q1
and Q3.
Q1:
𝜋
6
To find the Q3 angle, we simply
add pi to our reference angle.
𝜋
7𝜋
6
6
Q3: 𝜋 + =
EX: Use a calculator to
approximate arctan (4.84)
*Recall, tangent is positive in Q1
and Q3
Q1: 1.367
To find the Q3 angle, simply add
pi.
Q3: 3.14+1.367=4.509
EX: Use a calculator to
approximate arcos(-0.349)
Cosine is (-) in Q2 and Q3. Our
calculator gives the smallest
angle possible.
Q2: 1.927
The reference angle is
pi-1.927=1.214
To find the Q3 solution, add pi to
our reference angle.
Q3: pi+1.214=4.356
EX: Use a calculator to
approximate 𝑠𝑖𝑛−1 (−1.1)
No Solution – we can NOT take
the arcsine of a value greater
than 1 or less than -1.
EX: Find the exact value of
tan(arctan(-14)).
-14
EX: sin(arcsin(pi))
No Solution
EX: cos[arcos(0.54)]
0.54
EX: The angle of depression from
a lighthouse to a boat is 36
degrees. The boat is 300 feet
from the lighthouse. How tall is
the lighthouse?
(Draw a picture and show all
work on the board)
About 218 feet
EX: Find the exact value of
cos[arctan(-3/4)]
(show how to find the solution
both algebraically with a triangle
on with a calculator on the
whiteboard)
4/5
EX: Find the exact value of
sin[arcos(2/3)]
(show work on the whiteboard)
√5
3
A problem from Calculus:
EX: Write sec(arctan(x)) as an
algebraic expression in x.
Begin by letting u=arctan(x). This
means that tan(u)=x. Thus
tan 𝑢 =
𝑜𝑝𝑝
𝑎𝑑𝑗
𝑥
= .
1
(draw this triangle on the
whiteboard)
Therefore, the hypotenuse is
√1 +
𝑥 2.
Since 𝑠𝑒𝑐𝑎𝑛𝑡 =
ℎ𝑦𝑝
𝑎𝑑𝑗
,
our solution is √1 + 𝑥 2 .
*Stand up! Let’s review our
memorized trigonometric values
and identities once again.
Upon completion of this lesson,
you should be able to:
1. Find exact values for
arcfunctions.
2. Recite the unit circle.
For more information, visit
http://tutorial.math.lamar.edu/Extras/AlgebraTrigReview
/InverseTrig.aspx
HW Pg.349 6-84 6ths, 91-92, 94,
96, 105, 119