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Pre- Calculus 12
Date: _____________
Chapter 4– Trigonometry & the Unit Circle
Name: ____________________
Section 4.3B – Trigonometry Ratios (Primary& Secondary)
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Relating the trigonometric ratios to the coordinates of points on the unit circle
Determining exact and approximate values for trigonometric ratios
Identifying the measures of angles that generate specific trigonometric ratios
Solving problems using trigonometric ratios
For the following right-angled triangle,
AB is the _____________________ of the right triangle.
BC is _____________________ to angle θ.
AC is _____________________ to angle θ.
The following are called the primary trigonometric ratios: we use SOH CAH TOA.
opp
adj
opp
sin  
cos  
tan 
hyp
hyp
adj
The following are called the secondary trigonometric ratios. They are the _____________________
of the primary trigonometric ratios.
COSECANT
SECANT
COTANGENT
csc θ =
sec θ =
cot θ =
Example 1:
The point (4, -3) is on the terminal arm of an angle θ.
a) Draw θ in its standard position.
c) Find all six trig ratios in exact values.
sin θ =
csc θ =
cos θ =
sec θ =
tan θ =
cot θ =
b) Calculate θ.
Each quadrant gives us different signs:
ASTC
Q II
QI
Q III Q IV
Given the following conditions, where would  be?
a) cos  > 0 & tan  < 0
b) sin  < 0 & cot  > 0
Example 2:
Example 3:
Find sec θ in exact value if θ is in
5
quadrant II and cot θ =  .
12
Ex. 4
If θ is on a terminal arm, find the
measure(s) of angle θ if sin θ = 
1
.
5
Example 4: Determine each of the trigonometric ratios. Draw a diagram to support your answer.
a) cos 260 (nearest hundredth)
b) csc ( 70 ) (nearest hundredth)
c) sin 135 (exact value)
d) cot
5
6
Example 5: Determine the measures of all angles that satisfy the following. Use diagrams to support your
explanations.
a) cos  0.366, 0    360
c) Sin Ѳ = 0.879 , 0 ≤ Ѳ ≤ 2π
b) sec  
d)
2
, 0    2 .
3
To find any trigonometric ratio given an angle:
Step
1
Approximation
Exact Value
Set calculator to appropriate mode (degree
Determine the reference angle  R and which
or radian)
quadrant it lies in. Draw the right triangle to
the x-axis.
2
Enter trig ratio into calculator
Use the special triangles to determine the side
lengths of your triangle (the hypotenuse is
always 1 in the unit circle!)
Fill in the trig ratio including the sides (from
3
step 2) and the sign (from step 1).
To find an angle given any trigonometric ratio:
Step
1
2
Approximation
Exact Value
Determine all of the possible quadrants in
Determine all of the possible quadrants in
which our angle will lie (use ASTC). Draw
which our angle will lie (use ASTC). Draw a
a picture.
picture.
Enter inverse trig ratio (-1 exponent) into
Determine the reference angle  R using
calculator (check mode!) to find the
special triangles.
reference angle. Do not include the sign
of the ratio if it is negative!
3
a.
Use your diagram and your reference angle
Use your diagram and your reference angle to
to find all possible angles.
find all possible angles.
Sin Ѳ = 0.879 , 0 ≤ Ѳ ≤ 2∏
b.
Assignment: Page 201 #1-3, 5, 6, 8-10 (every other letter for all), + 12a, 14