6.3 Trig Functions of Angles
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Transcript of 6.3 Trig Functions of Angles
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6.3 TRIG FUNCTIONS
OF ANGLES
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So P = (x, y) is a point on the terminal side of theta. Using the Pythagorean theorem, r =
O
P
Q
opposite
adjacent
hypo
tenu
se
O
P (x, y)
Q
r
x
y
2 2x y
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So:
sin cos tan =
csc sec = cot =
0 x 0 y 0
y x yr r x
r r xy x y
y
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ExLet (-8, -6) be a point on the terminal side of theta. Find all six trig functions of theta.
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Quadrantal Angels – angels that are coterminal with the coordinate axis. When the terminal side of an angle theta, that is in
standard position , lies on the coordinate axes. (The angles for which either the x- or y-coordinate of a point on the terminal side of the angle is 0.)
Since division by 0 is undefined, certain trig functions are not defined for certain angles. EX is tan 90º degrees = y/x because x = 0.
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Evaluating Trig Functions at Any Angle ‘r’ is always positive because it is the
distance from the origin to the point P(x, y). The position of the terminal side will determine whether the trig function is positive or negative.
Q I: all positive Q2: (-x, y): sin, csc are + Q3 (-x, -y): tan, cot are + Q4 (-x, -y): cos, sec are +
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EX sin (-180º)
3tan sec 42
cos 270 csc2
cot( 90 )
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We see that the trig functions for angles that aren’t acute have the same value except possibly for the sign, as the corresponding trig functions of an acute angle. Let theta be an angle in standard position. The reference angle (theta bar) associated with theta is the acute angle formed by the terminal side of theta and the x-axis.
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ExFind the reference angle for:
30023
176
137
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ExFind the following: Cos 135º =
tan 390º
cos 60º
tan 240º
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Evaluating Trig Functions for any angle:1. Find the reference angle (theta bar)
associated with the angle theta.2. Determine the sign of the trig function of
theta by noting the quadrant in which theta lies.
3. The value of the trig function of theta is the same, except possibly for the sign, as the value of the trig function theta bar.
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Evaluate the Trig Functions using the Reference Angle.
15sin120 csc4
5cos120 tan3
7 5tan sin6 6
sec( 135 )
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Fundamental Identities
2 2
2 2
2 2
R eciprocal Identities:1 1 1csc sec = cot =
sin cos tansin costan cotcos sin
Pythagorean Identities:
sin cos 1
tan 1 sec
cot 1 csc
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Ex express sin in terms of cos
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Exexpress tan in terms of sin
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Ex
1tan and is in the fourth quadrant.2
Find cos
If
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Ex
2If tan and is in the third quadrant3
Find cos
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Ex If sec =6 and is in the fourth quadrant,find the other 5 trig functions of .
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Area of Triangles: With sides of length a & b and with included angle theta.
Find the area of triangle ABC:
1 sin2
A ab
120º
AB
C
18cm
3cm