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Section 2.1 - Angles in Standard Position
Initial arm – beginning arm (line) that lays on the x-axis
Terminal arm – line representing where the angle is located in a plane.
 ALWAYS COMES OUT OF THE ORIGIN
Angle in standard position – position of an angle when its
initial arm is on the positive x-axis and its vertex is at the origin
Reference angle – the acute angle between the terminal arm and the x-axis
Principal Angle – an angle between 0 degrees and 360 degrees
Example 1: Which angle is in standard position?
A
B
C
D
Quadrants
Degrees!!
2 (II)
1 (I)
3 (III)
4 (IV)
Quadrant 1 = 0⁰ - 90⁰
Quadrant 3 = 180⁰ - 270⁰
Quadrant 2 = 90⁰ - 180⁰
Quadrant 4 = 270⁰- 360⁰
Reference Angle Quick chart:
Q1: Reference Angle = Principal
Q2: Reference= 180⁰ - Principal
Q3: Reference= Principal - 180⁰
Q4: Reference = 360⁰ - Principal
Question 1: Find which quadrant each angle belongs in AND find the reference
angle for each of the following. CREAT A SKETCH OF THE DIAGRAM!!!
1) 150⁰
2) 218⁰
3) 272⁰
4) 52⁰ 5) 240⁰ 6) 145⁰ 7) 283⁰
Question 2: Find the principal angle for each of the following reference angles.
CREATE A SKETCH OF THE DIAGRAM!!!
a) 43⁰, Quadrant III
b) 72⁰, Quadrant IV c) 89⁰, Quadrant II d) 1⁰, Quadrant III
π or 3.1415…
WHAT IS AN EXACT VALUE?…
∘ answers involving no rounding or cutting off values
Special Right Triangles – (30⁰ -- 45⁰-- 60⁰)
√3
2
or 0.8660254038…
30⁰ Triangles
60⁰ Triangles
45⁰ Triangles
Example: Mr. H plays the piano. He uses a metronome to help keep time. The pendulum
arm of the metronome is 10cm long. For one particular tempo, the setting results in the
arm moving back and forth from a start position of 60 degrees to 120 degrees. What
horizontal distance does the tip of the arm move in one beat? Give an exact answer!!
HOMEWORK Page 83 #1-9, 11, 13, 17, 21
Section 2.2 Trigonometric Ratios of Any Angle
y
Quadrant II (90⁰<ɵ<180⁰)
Quadrant I (0⁰<ɵ<90⁰)
x
Quadrant III (180⁰<ɵ<270⁰)
Quadrant IV (270⁰<ɵ<360⁰)
Example 1: The point P(-5, -12) lies on the terminal arm of an angle, ɵ, in
standard position. Determine the exact trigonometric ratios for sin ɵ, cos ɵ,
and tan ɵ.
Example 2: The point P (2, -5) is on the terminal arm of an angle, ɵ, in
standard position. Determine the exact trigonometric ratios and determine
the measure of angle ɵ, to the nearest degree.
Example 3: State the quadrant(s) in which the terminal arm will be found. Then find the
other two trig ratios.
a)𝑠𝑖𝑛ɵ =
3
b) cosɵ =
4
−5
7
Example 4: Determine the exact value of sin240⁰
Example 5: Suppose ɵ is an angle in standard position with terminal arm in
1
quadrant III, and tanɵ = . Determine the exact values of sinɵ and cosɵ.
5
Quadrantal Angle
An angle in standard position
whose terminal arm lies on one
of the axes
UNDEFINED?!?!?! WHAT IS THAT??? We can cut a chocolate bar in
1
2
1
, but ??
0
Example 6: Determine the values of sinɵ, cosɵ, tanɵ when the terminal arm of
Quadrantal angle ɵ coincides with the negative x-axis, ɵ = 90⁰.
Example 7: Solve for ɵ.
a) sin ɵ = −0.5
b) cos ɵ =
√3
2
Example 8: Determine the measure of ɵ, to the nearest degree, given sin ɵ =
−0.8090, where 0⁰ ≤ ɵ ≤ 360⁰.
Extra Practice Questions
1) The point P(-5,7) is on the terminal arm of an angle in standard position.
a) find all the trig ratios using exact values
b) find the angle to the nearest tenth of a degree.
2) The point P(-6,-5) is on the terminal arm of an angle in standard position.
a) find all the trig ratios using exact values
b) find the angle to the nearest tenth of a degree.
3) State the quadrant(s) in which the terminal arm will be found. Then find the possible
values for angle. sinɵ =
3
5
4) Suppose ɵ is an angle in standard position with terminal arm in quadrant III, and sinɵ =
−3
4
. Determine the exact values of tanɵ and cosɵ.
5) Determine the values of sinɵ, cosɵ, and tanɵ for Quadrantal angles of 0⁰, 90⁰, 180⁰, and
270⁰.
Organize your answers in a table as shown below
0⁰
90⁰
180⁰
sinɵ
cosɵ
tanɵ
6) Solve sin ɵ = −
√2
2
, 0⁰ ≤ ɵ ≤ 360⁰
7) Determine the measure of ɵ, to the nearest tenth of a degree, given
cos ɵ = −0.6753, where 0⁰ ≤ ɵ ≤ 360⁰.
HOMEWORK Page 96 # 4-9, 11-13, 16, 18, 29
Means
perpendicular
Section 2.3 – The Sine LAW
270⁰
a)
b)
c)
D)
____________________________________________________________________
The sine law is a relationship between the sides and angles in any triangle.
C
𝑎
a
B
𝑠𝑖𝑛𝐴
b
c
=
𝑏
𝑠𝑖𝑛𝐵
=
𝑐
𝑠𝑖𝑛𝐶
OR
𝑠𝑖𝑛𝐴
𝑎
=
𝑠𝑖𝑛𝐵
𝑏
=
𝑠𝑖𝑛𝐶
𝑐
A
QUESTION: When is the best time to use the sine law?
ANSWER: When you have values for any pair - a side and its opposite angle.
Proof:
Example 1: Solve for the unknown side or angle.
a)
𝑎
𝑠𝑖𝑛42⁰
=
50
𝑠𝑖𝑛84⁰
b)
𝑠𝑖𝑛ɵ
15
=
𝑠𝑖𝑛98⁰
17.5
Example 2: Determine the length of all three sides and the measure of all three
angles (solve the triangle!!!)
21cm
48⁰
61⁰
Example 3: Solve the triangle
A
21cm
32cm
CHALLENGE QUESTION
In triangle PQR, angle P =36⁰, p=24.8m, and q = 23.4m. Determine the measure
of angle R, to the nearest degree!
Questions
1) Solve for x
2) Solve for sinɵ
𝑥
𝑠𝑖𝑛54⁰
=
45
𝑠𝑖𝑛80⁰
𝑠𝑖𝑛ɵ
𝑠𝑖𝑛85⁰
12
=
19.2
3) Solve the triangle if 24cm is opposite to 62⁰ and another degree of the triangle
is 56⁰
4) Solve the triangle if two lengths are given, 12m and 13m, with 57⁰ being
opposite of the 12 meter length
5) Calculate the perimeter of the triangle. One side length of a triangle is 17cm,
with 23⁰ being opposite to it. Another angle of the triangle is 80⁰.
Ambiguous Case
– from the given information the solution for the triangle is not clear…there might be
a) NO solution
b) one solution
c) two solutions
a < b sin A : NO SOLUTION
a = b sinA OR a ≥ b: ONE SOLUTION
b > a > b sinA: TWO SOLUTIONS
Example: In triangle ABC, angle A = 30⁰, a = 24cm, and b = 42cm. Determine the
measures of the other side and angles (round your answer to the nearest unit).
Example In triangle ABC, angle A = 39⁰, a= 14cm, and b = 10cm. Determine the measures of
the other side and angles. Express your answers to the nearest unit.
HOMEWORK page 108 # 1 – 9, 11, 12, 15, 21, 24
Section 2.4 The cosine law
If a, b, c, are the sides of a triangle and C is the angle opposite c, the cosine law is
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 𝑐𝑜𝑠𝐶
QUESTION: When do you use the cosine law?
ANSWER: If the triangle has 2 given sides and the included angle or the triangle
has 3 given sides and no angles.
Example 1: Calculate the value of the unknown side, x.
x
17
14
10
39⁰
78⁰
Example 2: Calculate the value of the unknown side, x.
12
x
Example 3: Calculate the value of the unknown side, x.
28⁰
22
x
14
Example 4: Solve the triangle
14
19
23
Questions
1) Determine the length of BC to the nearest tenth of a centimeter.
a)
C
b) C
15cm
44⁰
A 19cm
144⁰
15cm
B
B
A
21cm
2) In triangle DEF, DF=18cm, DE=12cm, and EF = 10cm. Determine the
measure of angle F to the nearest degree.
3) Nina wants to find the distance between two points, A and B, on opposite
sides of a pond. She locates a point C that is 35.5 m from A and 48.8 from B. If
the angle at C is 54⁰, determine the distance AB, to the nearest tenth of a meter.
4) In triangle ABC, a=9, b = 7, and angle C = 33.6⁰. Sketch a diagram and
determine the length of the unknown side and the measures of the unknown angles,
to the nearest tenth.
5) The distance from tee to hole on a golf course is 290 yards. John hit his first
shot 150 off target to the left a distance of 170 yards. How far will his ball be from
the hole for his second shot?
6) A triangular brace has side lengths 12m, 18m, and 22m. Determine the
measure of the angle opposite the 18-m side, the nearest degree.
HOMEWORK Page 119 1 a,c
2 b,d 3, 4 a,c,f
5, 10, 11, 13, 19, 21