5.3: Trigonometric Ratios of Obtuse Angles
Unit 5: Trigonometric Ratios
MCR3U1: Functions
http://www.dpcdsb.org/AMBRO
Action - Terminology
Angles and Their Location in the x-y Plane
y
<<
<<
x
<<
<<






An angle is formed when a ray is rotated about a fixed point called the
.
The ray is called the
at the beginning of the angle and the
at the end of the angle.
Angles are often labelled with Greek letters, such as  ”theta,”  “alpha,” and  “beta”.
Angle  is called the
An angle,  is in
if :
 the vertex of the angle is at the origin,
 the initial arm lies fixed along the positive x-axis, and
 the terminal arm is the final position of the rotating ray anywhere on the arc of
rotation.
Angle β is called the
:
Angle Direction

An angle can have a positive or negative value:
y
y
x
x
A
angle is formed by a
counter-clockwise rotation of the terminal arm.
A
angle is formed by a
clockwise rotation of the terminal arm.
Action – Relationships With Angles and Ratios
Fill out the table below by finding the ratio of each angle using a calculator. Make sure your
calculator is in degree mode. Round to 3 decimal places.
Angle
(Degree)
0
15
30
45
60
75
90
105
120
135
150
165
180
195
210
225
240
255
270
285
300
315
330
345
360
Quadrant
sinA
cosA
tanA
1. Examine all the sine ratios that are identical. What is the relationship between these
angles?
2. Examine all the cosine ratios that are identical but have opposite signs. What is the
relationship between these angles?
3. Examine all the tangent ratios that are identical but have opposite signs. What is the
relationship between these angles?
4. What other relationships do you observe?
Primary Trigonometric Ratios



Let point P(x, y) be a point on the terminal arm of angle  in standard position.
The side opposite  is y and the side adjacent to  is x.
We choose r as the hypotenuse since it represents the radius of a circle.
y
P(x, y)

x
PRIMARY TRIGONOMETRIC ratios can be defined in terms of
the x and y co-ordinates of any point P(x, y):
If
r=
then,
1. sin  =
2. cos  =
3. tan  =
The “CAST” Rule
The “CAST” Rule tells us which trig ratios are POSITIVE in each quadrant.
Investigation
Complete the table below to find the SIGN of each
primary trig ratio for the angle in each quadrant:
QUADRANT
y
r
x
cos  
r
y
tan  
x
sin  
Which ratio is
Positive?
I
II
III
II
I
III
IV
IV
Ex. 1: The point (–6, 8) is on the terminal arm of an angle  in standard position.
Find sin  , cos  , and tan . What is the measure of  ?
Ex. 2: Find angle C to the nearest degree, if 0    360.
a) sin C = 0.4226
b) cos C = -
4
5
Coterminal Angles

Coterminal angles are angles in standard position that have the same TERMINAL arm.
Ex.3: State two coterminal angles for a) 30 and for b) 120 .