1.4 Definition of the Trigonometric Functions
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Transcript of 1.4 Definition of the Trigonometric Functions
1.4 Definition of the Trigonometric FunctionsOBJ: Find the values of the six trigonometric functions of an angle in standard position
DEF: Standard Position
Angle with its vertex
at the origin,
initial side on the x-axis,
and terminal side
containing a point P
with coordinates (x,y)
P (x,y) y
x
5
5
-5-5
Draw in standard position.
Choose any point P
with coordinates (x, y)
on the terminal side of
(P cannot be the
vertex of ) P (x,y)
y
x
5
5
-5-5
y
x
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Draw the to the x-axis at Q, forming ∆OPQ. PO = r > 0.
x2 + y2 = r2
P
Q O
DEF: Trigonometric Functions
X2 + Y2 = R2
P
(Opp) Y R (Hyp)
(Adj) X
SOHCAHTOASYRCXRTYX
Sin = Y Opp Csc = R Hyp R Hyp Y
Opp
Cos = X Adj Sec = R Hyp
R Hyp X Adj
Tan = Y Opp Cot = X Adj X Adj Y Opp
y
x
5
5
-5-5
All Students Take Calculus
S A
T C
Sin All
Tan Cos
y
x
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-5-5
y
x
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-5-5
EX: 1 The terminal side of an angle in standard position goes through the point ( 8 , 15 ). Find the values of the six trigonometric functions of P 35 (8, 15)
Sin = Y 15 Csc = R 17 R 17 Y 15
Cos = X 8 Sec = R 17 R 17 X 8 15
Tan = Y 15 Cot = X 8 X 8 Y 15
8
y
x
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-5-5
y
x
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-5-5
EX 2 The terminal side of an angle in standard position goes through the point (-3, - 4 ). Find the values of the six trigonometric functions of .P 36
Sin = Y = -4 Csc = R = 5 R 5 Y -4 Cos = X = -3 Sec = R = 5
R 5 X -3Tan = Y = -4=4 Cot = X = -3 3 -3 X -3 3 Y -4 4 -4
(-3,-4)
y
x
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-5-5
EX 4 Find the values of the six trigonometric functions for an angle of 90 .P 39Sin = Y = 1 Csc = R = 1
R 1 Y 1 (0, 1)
Cos = X = 0 Sec = R = 1 = Ø
R 1 X 0
Tan = Y = 1 = Ø Cot = X = 0 X 0 Y 1
y
x
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DEF: Quadrantal angles
TABLE OF VALUES OF THE SIX TRIGONOMETRIC FUNCTIONS OF QUAD. S
0
90
180
270
360
450
DEF: Coterminal Angles
Angles with the same initial side and
the same terminal side
EX: Find the angles of smallest possible measure coterminal
with the following angles: 900
y
x
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EX: Find the angles of smallest possible measure coterminal
with the following angles: 900 900 – 720 180
y
x
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DEF: Quadrantal angles
TABLE OF VALUES OF THE SIX TRIGONOMETRIC FUNCTIONS OF QUAD. S
Sin Cos Tan Cot Sec Csc
0 0 1 0 Ø 1 Ø
90 1 0 Ø 0 Ø 1
180 0 -1 0 Ø -1 Ø
270 -1 0 Ø 0 Ø -1
360 0 1 0 Ø 1 Ø
450 1 0 Ø 0 Ø 1