MCR 3U – Unit 5: Trigonometry
Date: __________________
The CAST Rule
Recall:
To work with angles greater than 90° , we formed a right-triangle using the terminal arm and the related
acute angle. (Lesson 3)
For angles in Q1 and Q2,
sin θ = sin(180° − θ )
cos θ = − cos(180 ° − θ )
tan θ = − tan(180° − θ )
Let’s expand this list into Q3 and Q4.
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The CAST Rule
Any angle in standard position has a related acute angle. An acute right-triangle can always be drawn
using this RAA, thus any angle can be associated with the primary trig ratios.
The quadrant will determine the sign (+ or -) of the ratio.
The CAST rule allows us to quickly determine the sign of each trig ratio for any quadrant.
Since tangent is
positive in Q3, we can
add to the list that
tan θ = tan(180° + θ )
sin θ = − sin(180° + θ )
cos θ = − cos(180° + θ )
Since cosine is
positive in Q4, we can
add to the list that
cos θ = cos(360 ° − θ )
sin θ = − sin(360° − θ )
tan θ = − tan(360° − θ )
Ex. 1: Predict the sign (+ or -) of each value. (Verify using your calculator)
a) tan 135°
b) cos 240°
c) sin 430°
d) tan(−30°)
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5
, where 0° ≤ θ ≤ 360°
24
a) In which quadrant(s) could the terminal arm of θ be located?
Ex. 2: For tan θ = −
b) Determine all possible trigonometric ratios for θ .
c) Evaluate all possible values of θ to the nearest degree.
Practice Questions:
1. Given angle θ , where 0° ≤ θ ≤ 360° , determine two possible values of θ where each ratio would be
true. Sketch both principal angles.
a) cos θ = 0.6951
b) tan θ = −0.7571
c) sin θ = 0.3154
d) cos θ = −0.2882
e) sin θ = −0.7503
5
, where 0° ≤ θ ≤ 360°
12
a) In which quadrant(s) could the terminal arm of θ be located?
b) Determine all possible trigonometric ratios for θ .
c) Evaluate all possible values of θ to the nearest degree.
2. For cos θ = −
Answers to Practice Questions:
1.
2. a) Quadrant 2 and 3.
119
5
119
, cos θ = −
and tan θ = −
.
12
12
5
− 119
5
119
In Quadrant 3: sin θ =
, cos θ = −
and tan θ =
.
12
12
5
c) 115° and 245°
b) In Quadrant 2: sin θ =
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