Trigonometric Ratios
- ratios used to find unknown side lengths or angle measures in RIGHT triangles.
Recall: Primary Trig Ratios
sinA =
cosA =
tanA =
Angle of Elevation
- the angle measured ____________ between the horizontal and the line of sight
from an observer to an object.
Angle of Depression
- the angle measured ______________ between the horizontal and the line of sight
from an observer to an object.
Ex. 1) Solve ΔQRS, if Q = 90o, R = 42o and q = 10.5 cm.
Ex. 2) Find θ to the nearest degree.
Ex. 3) Two office buildings are 42.3 m apart. From the top of the shorter one, the
angle of elevation to the top of the other is 27.5o, while the angle of depression to
the base is 78.2o. Find the height of each building.
Sine and Cosine Laws
Recall from last year:
SINE LAW
COSINE LAW
When do we use:
SINE LAW
COSINE LAW
Ex. 1) Determine the value of x.
Ex. 2) Solve Δ PQR.
Ex. 3) Find the measure of angle A.
Practice:
pg. 290 # 1 - 4 (c & d only for each), 5d
Applications of Trigonometry
Ex. 1) An escalator rises 8m vertically and makes an angle of 25o with the
horizontal. What is the length of the escalator to one decimal place?
Ex. 2) A surveyor in a helicopter, 9750m directly above ground observes that the
angle of depression to two points on the opposite shores of a lake are 32o and 45o,
respectively as shown in the diagram. What is the width of the lake (between the
two points), to the nearest meter?
Sine Law - The Ambiguous Case
A weather balloon is tethered by two ropes; one is 7.8m long, which makes an angle of
36o with the ground. The other rope is 5.9m long. Determine the distance between the
two ropes.
* from the problem, it is not clear how Albert & Belle & the balloon are positioned
relative to one another.
CASE 2: Assume A & B are on the same side of the balloon.
When does the ambiguous case occur?
Given two sides and a non-contained angle, the triangle may NOT be unique.
(SSA)
A) Given angle A , if a > b, one triangle exists
B) Given angle A, if a < b, then
Number of triangles
Condition
Ex. 1) How many triangles exist in each case?
) M = 42o
m = 9.6cm
n = 6.5cm
Ex. 2) Solve Δ XYZ, if:
X = 33o
x = 4.1cm
y = 5.4cm
b) P = 39o
p = 5.6m
q = 10.5m
c) K = 48o
k = 7.5mm
l = 8.1mm
AMBIGUOUS CHECK-UP:
Solve the following triangle.
𝛥𝐴𝐵𝐶 𝑔𝑖𝑣𝑒𝑛∠𝐴 = 12𝑜 , 𝑎 = 32 𝑚 𝑎𝑛𝑑 𝑏 = 45 𝑚
FIRST identify: How many triangles will we have?
Step 1: Use the sine law to calculate B
Step 2: Determine ACB
(all angles of a triangle add to 180
degrees)
ACB  ______ 
Step 3: Find the length of AB (using the
sine law).
B  ______ 
AB = ____________ m
Consider AB' C (the one triangle on the left!)
Step 4: Seeing that you have an isosceles triangle, calculate AB' C (marked as z)
z  ______ 
Step 5: Calculate ACB' (all angles of a triangle add to 180 degrees)
Step 6: Calculate the length of AB’, using the sine law.
Angles in Standard Position
Definitions:
Let P(x, y) be any point on the
terminal arm of angle θ in standard
position, r units from the origin.
Re-write the primary trig ratios:
sin θ =
cos θ =
tan θ =
How can the measure of r be found?
IF the terminal arm lies in quadrant I,
then θ is the value found using the
previous trig ratios.
IF the terminal arm lies in quadrants II, III, or IV, there is a principal angle θ, and a RELATED
ACUTE ANGLE (RAA).
Ex. 1) Find the sinθ, cosθ, tanθ and principal angle θ given the point P is on the terminal arm of
an angle θ in standard position.
Steps:
a) P(7, 24)
1. Decide which quadrant the
terminal arm lies in based
on the point given.
2. Draw a diagram with the
terminal arm in the correct
quadrant.
3. Determine the value of r.
4. Determine the ratios.
5. Use the ratios to determine
the principal angle or RAA.
b) P(-3, -9)
c) P(10, -5)
Practice:pg. 348 #1abef + principal angle for all
#2abefgh + principal angle for all
Treat 0 ≤ θ≤ 2π as 0o ≤ θ≤ 360o
Correction: 1a) cosθ = 8
17
C.A.S.T. Rule
Can you tell what the following all have in common?
sin 35o
sin 145o
sin 215o
sin 325o
Evaluate:
sin 35o
sin 325o
sin 145o
sin 215o
Quadrant II
90o ≤ θ ≤ 180o
sinθ =
cosθ =
tanθ = III
Quadrant
180o ≤ θ ≤ 270o
Quadrant I
0o ≤ θ ≤ 90o
sinθ =
cosθ =
tanθ =
sinθ =
cosθ =
tanθ =
C.A.S.T. Rule
Quadrant IV
180o ≤ θ ≤ 360o
- tells us where each trig ratio is positive
sinθ =
cosθ =
tanθ =
We use the CAST rule to determine the
correct sign of any trig ratio.
Ex. 1) Use the CAST rule to determine the two angles (nearest tenth) associated with cos θ = 0.627
Steps:
between 0o ≤ θ ≤ 360o
1. Use CAST to determine
which
quadrants the terminal arm
could lie in.
2. Draw diagram including
labels.
3. Find the RAA - by taking the
inverse of the POSITIVE
Ex. 2) Find the two angles 0o ≤ θ ≤ 360o associated with the followingratio
trig ratio.
tan θ = - 0.2679
value.
4. Use RAA to find both
possible
angles of the terminal arm.
Ex. 3) If θ is in standard position, with its terminal arm in the specified quadrant and 0o ≤ θ ≤
360o, find the exact value of the other two trig ratios and the principal angle.
cos θ = 4 , quadrant IV
7
Practice:
pg. 349 # 6, 7
0o ≤ θ ≤ 360o for all questions
Warm up
Find <A, to the nearest degree if 0o ≤ θ ≤ 360o. Draw a diagram to support your answers.
a) tan A = -1.0355
b) sin A = 0.5299
Special Angles / Triangles
Most trig ratios cannot be expressed exactly. For example, sin25o = 0.422618262... is irrational
However, some trig ratios can be expressed exactly...
Special Triangle #1
sin 45 =
cos 45 =
tan 45 =
Special Triangle #2
sin 30 =
cos 30 =
tan 30 =
Ex. 1) Find the exact value of each trig ratio.
Steps:
1. Draw terminal arm on
diagram.
2. Determine RAA.
3. Label the special triangle.
4. Find the value required. (use
CAST to help determine the
sign)
o
a) sin 60
b) cos 225o
c) tan 330o cos 135o
Ex. 2) If 0o ≤ θ ≤ 360o, find the possible answers of angle θ.
a) sinθ = -1
b) tanθ = -√3
√2
Pg. 348 - 350#3 (include sketches), 11, 18
Pg 353 #1 ab (sine only), #2 ac (cosine only), #4 ab
No calculators permitted. Diagrams required.
1.  Evaluate.
a) cos 60o
b) sin 150o
c) sin 150o
d) sin 225o
e) tan 135o
f) sin 180o
g) cos 90o
180o
h) sin 270o
i) sin 360o
j) tan
No calculators permitted. Diagrams required.
Given, 0    360, calculate the value of 
a.) cos  
3
2
b.) tan   3
c.) sin   0
Review of Unit 5
A. Solve the following triangle
ABC (C = 35, c = 11 m, a = 12 m)
B. Angles in Standard Position
1. If the point ( – 2, – 4) lies on the terminal arm of angle,  , in standard
position:
i) Draw a labelled sketch of the terminal arm.
ii) Determine the value of r .
iii) Determine the value of the related acute angle  .
iv) Determine the value of the principal angle  .
2. Solve for angle  , where 0     360  .
a) tan   0.34432
b) sin   0.40674
3. Calculate the exact value of the following:
a) tan 330o  x
b) sin120o  x