This document summarizes key concepts from a chapter on trigonometric functions. It defines the six trig functions (sine, cosine, tangent, cosecant, secant, cotangent) in terms of right triangles and provides examples of how to use trig functions to find unknown side lengths. It also introduces special right triangles used to find exact values of trig functions, such as the 45-45-90 triangle used to find sine and cosine of 45 degrees.

1. Chapter 4: Trigonometric Functions
Section 4.3 (pg. 277-287)

2. • If we want to convert degrees to radians we multiply by…
π
180
• If we want to convert radians to degrees we multiply by…
180
π
A degree is an angle
measure, a radian is an
arc measure

3. What is the Pythagorean Theorem?
• In a right triangle, the length of one leg squared plus the length of
the second leg squared equals the squared length of the
hypotenuse.
So what does that mean in English?
a2 + b2 = c2

4. - A right triangle is a triangle with a
right angle. Let’s look at one of the
acute angles of the triangle, which we
will call theta = θ
θ
- The side opposite the right angle is
called the hypotenuse
- The side opposite θ is called the
opposite side (makes sense right?)
- The side next to θ is called the
adjacent sideAdjacent
Opposite
Using the lengths of these 3 sides,
you can form six ratios that define
the six trigonometric functions of
the acute angle θ

5. sin(x)
cos(x)
tan(x)

6. cosecant(x)
secant(x)
cotangent(x)

7. - If θ is the acute angle of a right
triangle, then we can find the
values of sin(θ), cos(θ) and tan(θ)
using the side lengths of the
triangle like so:
sin(θ) =
cos(θ) =
tan(θ) =
opposite
opposite
hypotenuse
hypotenuse
adjacent
adjacent

8. θ
Find:
sin(θ) = =
cos(θ) = =
tan(θ) = =
opposite
hypotenuse
opposite
adjacent
adjacent
hypotenuse
4
5
3
5
4
3

9. I
N
E
P
P
O
S
I
T
E
P
P
O
S
I
T
E
Y
P
O
T
E
N
U
S
E
Y
P
O
T
E
N
U
S
E
O
S
I
N
E
A
N
G
E
N
T
D
J
A
C
E
N
T
D
J
A
C
E
N
T

10. - If θ is the acute angle of a right
triangle, then we can also find the
values of cosecant(θ), secant(θ) and
cotangent(θ) using the side lengths
of the triangle like so:
cosecant(θ) =
secant(θ) =
cotangent(θ) =
hypotenuse
adjacent
opposite
adjacent
opposite
hypotenuse

11. θ
Find:
csc(θ) = =
sec(θ) = =
cot(θ) = =
hypotenuse
opposite
adjacent
opposite
hypotenuse
adjacent
5
4
5
3
3
4

13. Take a few minutes and make some conjectures
with your table group about the relationships
between the 6 trig functions
Write down at least one conjecture

15. - In the earlier example, you were given the side lengths of a triangle
and asked to find sin(θ), cos(θ), and tan(θ). Sometimes you will be
asked to find sin, cos, tan for a given acute angle, like θ = 45⁰
- How can you find the value of sin(45⁰)?
• We could use a calculator: sin(45⁰) = .7071067812
• But that’s just an approximation, what if we wanted the
exact value, not the decimal approximation?
We can construct a right triangle with
45⁰ as one of its acute angles

16. Find the exact value of sin(45⁰)
• We will start by drawing a triangle with 45⁰ as
one of its acute angles. The length of the adjacent
side will be 1 unit.
1
45⁰
• Now we can use a little bit of
Geometry to help us out. Since two
angles of the triangle are 90⁰ and 45⁰,
the other angle has to be … 45⁰
45⁰
• Since the base angles are congruent,
the legs must be congruent also, so
the other side length is …
1
1
• Finally, the Pythagorean
Theorem tells us that the
hypotenuse is …
√2
√2

17. Find the exact value of sin(45⁰)
1
45⁰
45⁰
1
√2
• Now we are almost done! We can use SOH CAH TOA
to find sin(45⁰)
sin(45⁰) = =
opp
hyp
1
√2
Now find cos(45⁰) and tan(45⁰)
and check your answer with a
calculator
Use your calculator to make sure that
this is the answer we got earlier

18. 1
45⁰
45⁰
1
√2
This triangle
has a special
name …
It is called a 45-45-90 right triangle, and we
use it to find sin(45⁰), cos(45⁰) and tan(45⁰)

19. 1
60⁰
30⁰
2
This triangle also has a special name
It is called a 30-60-90 right
triangle, and we can use it to find
trig values involving 30⁰ and 60⁰
√3
Find:
tan(60⁰)
cos(30⁰)
sin(30⁰)
cos(60⁰)
Now check your answers
with a calculator

21. • Pioneer’s Clock Tower is 40 ft high, and you are standing 40 ft away
staring at the top of the tower. At what angle is your head looking upward?
o Let’s draw a picture. We know you are 40 ft away and the tower is 40
ft high. We will call the angle your head is looking up θ.
θ
40
40
o We have the opposite and the adjacent
sides of the right triangle, so we can say:
tan(θ) = 40
40
= 1
And we know from our
45-45-90 triangle that
θ must be …
45⁰

22. Brendan Gibbons kicks the football at an angle of 60⁰ when he is 45 yards
away from the field goal post. How far did the football travel through the air?
60⁰
45 yards
o Since this is a right triangle, we can set up a trig equation. Should we
use sin(60⁰), cos(60⁰), or tan(60⁰)?
cos(60⁰) = adj
hyp
45
x
= o And now since cos(60⁰) = ½ we can say…
1 45
2 x
=
o And we solve for x …