Unit 3 – Polynomial Functions, Equations and...
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Unit 3 Polynomial Functions, Equations and Inequalities
Chapter 6 Trigonometry 28
29 Chapter 6 Trigonometry
q
Chapter 6 Trigonometry
Name: _______________
6.1 Trigonometric Ratios for Any Angle in Standard PositionDate:
Standard Position: An angle is in standard position when the initial arm starts at (0, 0) and lies on the positive xaxis; the terminal arm is rotated about (0, 0).
Angles in standard position are graphed on the same plane as functions, and we label the four sections as quadrants:
=
=
=
=
=
=
=
=
=
adjacent
side
opposite
side
tan
hypotenuse
adjacent
side
cos
hypotenuse
opposite
side
sin
q
q
q
yx1234223223221234
yx
yx
Example 1: For each angle, draw the angle in standard position:
a)
=
120
q
b)

=
85
q
yx
yx
Suppose the ray continues to rotate. When the ray first reaches its starting position it forms an angle of ______ (if rotating counterclockwise) and form an angle of ______ (if rotating clockwise). After rotating all around until a full cycle, you will end on the same position as standard position, this new angle will be either greater than 360 or less than 360.
Example 2: For each angle in example 1, determine the angle after a full cycle around.
When a ray rotates more than 1 revolution, it forms angles greater than 720 (when rotating counterclockwise) and form angles less than 720 (when rotating clockwise).
Example 3: For each angle in example 1, determine the angle after 2 full cycles around.
You can continuously adding or subtracting 360 and you will end up with the same terminal forever. These values are called coterminal angles.
Coterminal Angles: Two or more angles in standard position, when the position of P is the same for each angle. When represents any angle, then any angle coterminal with is represented by these expressions, where n is any integer.
n
360
q
, where
n
(natural number)
You TryIdentify the angles coterminal with 210 that satisfy the domain 720 720. Express the angles coterminal with 210 in general form.
Reference Angles
Definition: Consider the angle in standard position. The reference angle of is _____________________________________________________________________.
Sketch the reference angle, r, of each angle .
q
=
=
=
=
=
=
=
=
=
adjacent
side
opposite
side
tan
hypotenuse
adjacent
side
cos
hypotenuse
opposite
side
sin
q
q
q
yx551010510105
Unit Circle
The unit circle is simply a circle whose radius equals _______.
For our purposes, we will place the centre of a unit circle at the origin of the Cartesian plane.
Relating tan to sin and cos
Since
y
=
q
sin
and
x
=
q
cos
for the unit circle. Then
tan
=
=
x
y
q
Special Angles:
Recall the two special triangles:
Definition of Secant, Cosecant, and Cotangent
In mathematics, the ratios
q
cos
1
,
q
sin
1
,
q
tan
1
(reciprocal) are used so often that we give them special names.
We define the secant, cosecant, and cotangent functions are follows:
Secant of
Cosecant of
Cotangent of
Example 4: Determine the trigonometric ratio for each degree
in degrees
0
30
60
90
45
180
270
360
sin
cos
tan
sec
csc
cot
Example 5: Determine the exact value of the six trigonometric ratios of 495.
You Try Determine the exact value of the six trigonometric ratios of 150.
Determining Trigonometric Ratios of nonspecial triangles.
Example 6: Determine the exact value of the six trigonometric ratios of 200.
Determining All Trigonometric Ratios and Angle Measure given one Trigonometric Ratio.
Example 7: a) Suppose
13
12
cos
=
q
. Determine the exact values for the other trigonometric ratios for