UNIT6.53CIRCLE
STM IV
OBJECTIVES
• Recall how to determine
coordinates of a given
point.
• Determine coordinates of a
special angles using
special triangle
relationships.
• Determine coordinates of a
special angles using sine
and cosine functions.
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DETERMINE THE
COORDINATES
OF THE GIVEN
POINTS:
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REVIEW:
How do we determine
coordinates of a given
point?
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UNIT CIRCLE
Circle whose center point is
the origin, and has a radius
equals to 1 unit.
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TASK 1:
Draw terminal sides for angle
30, 45 and 60 degrees.
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DETERMINE
COORDINATES
USING SPECIAL
ANGLE
RELATIONSHIPS.
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30°
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(
7
3 1
, )
2 2
2 2
( , )
45°
2 2
DETERMINE
COORDINATES
USING SPECIAL
ANGLE
RELATIONSHIPS.
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1 3
( , )
60° 2 2
DETERMINE
COORDINATES
USING SPECIAL
ANGLE
RELATIONSHIPS.
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SPECIAL
ANGLES
•
30, 45, 60, 90 degrees are called special
angles.
•
Angles whose reference angles are 30, 45
and 60 are also considered as special angles.
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TASK 2:
Determine other special angles by creating
terminal side for angles whose reference
angles are 30, 45 and 60 in QII – QIV.
Determine the coordinates of the special
angles.
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NOTE:
Angle measures with the
same
reference
angles
have the same coordinate
values except for the sign.
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OBJECTIVES
• Determine coordinates of a
special angles using sine
and cosine functions.
• Determine values of 6
trigonometric functions.
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DETERMINE
COORDINATES
USING
TRIGONOMETRIC
RELATIONSHIPS.
Sin 𝛼
Cos 𝛼
In General, coordinates of a
point =
(Cos x, Sin x)
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TRIGONOMETRIC RATIOS:
Sin = Opp / Hyp
Csc = Hyp / Opp
Cos = Adj / Hyp
Sec = Hyp / Adj
Tan = Opp / Adj
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Cot = Adj / Opp
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RECIPROCAL IDENTITIES:
Sin = 1/CSC
Csc = 1/SIN
Cos = 1/SEC
Sec = 1/COS
Tan = 1/COT
Cot = 1/TAN
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OTHER TRIGONOMETRIC
IDENTITIES:
Pythagorean Identities Other Identities:
𝒔𝒊𝒏𝟐 𝒙 + 𝒄𝒐𝒔𝟐 𝒙 = 𝟏
𝟐
𝟐
𝒔𝒊𝒏 𝒙 = 𝟏 - 𝒄𝒐𝒔 𝒙
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Cot 𝜶 =
𝒄𝒐𝒔 𝜶
𝒔𝒊𝒏 𝜶
𝟐
𝒄𝒐𝒔 𝒙 = 𝟏 - 𝒔𝒊𝒏 𝒙
𝟐
Tan 𝜶 =
𝒔𝒊𝒏 𝜶
𝒄𝒐𝒔 𝜶
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VALUES OF 6 TRIG FUNCTIONS
Angle
measure
s in
Degrees
COS
SIN
TAN
SEC
CSC
COT
0
30
45
60
90
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