Notes #4-5: Trigonometry Extended: The Circular Functions (Section 4.3 - day 1)
I can: define initial side, terminal side, positive angles, negative angles, vertex,
standard position, conterminal angles
I can: use reference angles to evaluate trig functions in any quadrant.
In geometry we think of an angle as 2 rays that
meet.
Terminal Side (final)
In trigonometry we think of angle in terms of a
rotating ray.




The beginning position of the ray, the initial
side, is rotated about its endpoint called the
vertex.
The final position is called the terminal side.
The measure of an angle is the number that
describes the amount of rotation from the
initial side to the terminal side.
All angles start in Standard Position,
with the initial side on the positive 𝑥-axis
and vertex at the origin.
Measure of the angle –
amount of rotation from
initial (start) to terminal
(finish)
Vertex
Initial Side (start)

We can also rotate past 180° to create
much larger, obtuse angles than we have
worked with before.

We can rotate completely around the
vertex to create a 360° angle:

We can rotate more than once around
the vertex to create extremely large
angles:
START


Positive angles rotate
counterclockwise
Negative angles rotate clockwise
530°
−375°

Coterminal Angles – 2 angles that share the same initial and terminal side.
o Coterminal angles differ by multiple of 306° or 2𝜋.
o Coterminal angles can be positive/negative; negative/negative;
positive/positive
Example 1: Finding Coterminal Angles
Find and draw a positive angle and a negative angle that are coterminal with the given
angle
a) 55°
b) −160°
c)
3𝜋
4
Need more help? – see pg. 339 ex. 1
We can find the 6 trigonometric functions of any angle, even obtuse angles like 175°.
Remember, our trig functions from earlier in the chapter have only worked for acute angles;
today we are going to apply the trig functions to ALL ANGLES. We just need to be creative
with how we do the math!
sin 175° =
sin 5° =
cos 350° =
cos 10° =
tan 225° =
tan 45° =
All of the angles,
have the same
trig values! The 1st
column are
obtuse angles,
the 2nd column
are acute angles!
Reference Angles – an acute angle formed by the terminal side (final side) of the given
angle and the 𝒙-axis. Every angle has a reference angle. Look at the picture below. The
terminal side is blue, the 𝑥-axis is yellow.
We form reference angles to apply trig ratios too all angles (not just acute angles).
1st Quadrant
2nd Quadrant
3rd Quadrant
4th Quadrant
Example 2: Finding Reference Angles
Sketch the angle. Then determine its reference angle.
118°
199°
352°
12°
We use reference angles to find trig values of obtuse angles!
Example 3: evaluating trig functions determine by a point
Let 𝜃 be any angle in standard position whose terminal side contains the point (−4, 6). Find
the six trigonometric functions of 𝜃.
Let 𝜃 be any angle in standard position whose terminal side contains the point (2, −8). Find
the six trigonometric functions of 𝜃.
Need more help? – see pg. 340 ex. 2 & 3
DEFINITION Trigonometric Functions of ANY angle
Let 𝜃 be any angle in standard position.
Let Point (𝑥, 𝑦) be any point on the terminal side of the angle.
Let 𝑟 denote the distance from the origin to Point (𝑥, 𝑦).
𝒓
𝒚
𝐜𝐬𝐜 𝜽 = ; 𝒚 ≠ 𝟎
𝐬𝐢𝐧 𝜽 =
𝒚
𝒓
𝒙
𝒓
𝐜𝐨𝐬 𝜽 =
𝐬𝐞𝐜 𝜽 = ; 𝒙 ≠ 𝟎
𝒓
𝒙
𝒙
𝒚
𝐜𝐨𝐭 𝜽 = ; 𝒚 ≠ 𝟎
𝐭𝐚𝐧 𝜽 = ; 𝒙 ≠ 𝟎
𝒚
𝒙
𝜃
𝑥
𝑦
Reference angle
𝑟
𝑟 is the
terminal side
Pythagorean Theorem based on reference triangle: 𝒙𝟐 + 𝒚𝟐 = 𝒓𝟐
You may need to find 𝒓 !
From the previous definition, we can summarize some information:
0°/0/2𝜋
If point 𝑷 is on the terminal side of angle 𝜽, evaluate the 6 trigonometric functions for 𝜽
(𝟎, 𝟒)
(−𝟐, 𝟎)