Download Six Trigonometric Functions of t

Trigonometric Functions
Unit Circle Approach
The Unit Circle
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Definition
Six Trigonometric Functions of t
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The sine function associates with t the ycoordinate of P and is denoted by
sin t = y
The cosine function associates with t the xcoordinate of P and is denoted by
cos t = x
Six Trigonometric Functions of t
If x  0, the tangent function is defined as
y
tan t 
x
If y  0, the cosecant function is defined as
1
csc t 
y
Six Trigonometric Functions of t
If x  0, the secant function is defined as
1
sec t 
x
If y  0, the cotangent function is defined as
x
sec t 
y
Finding the Values on Unit Circle
Find the values of the six trig functions given the
point on the unit circle

5 2
,  
 
3
 3
Six Trigonometric Functions of the
Angle θ
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If θ = t radians, the functions are defined as:
sin θ = sin t
csc θ = csc t
cos θ = cos t
sec θ = sec t
tan θ = tan t
cot θ = cot t
Finding the Exact Values of the 6 Trig
Functions of Quadrant Angles
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Unit Circle – radius = 1
Quadrant Angles: 0, 90, 180, 270, 360 degrees
0, π/2, π, 3π/2, 2π
Point names at each angle:
(1, 0)
(0, 1)
(-1, 0)
(0, -1)
Finding the Exact Values of the 6 Trig
Functions of Quadrant Angles
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Table with all of values on p. 387
Circular Functions
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A circle has no beginning or ending. Angles
on a circle therefore have many names
because you can continue to go around the
circle.
Positive Angles
Negative Angles
Finding Exact Values
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Reminder of how to use your hand to find the
value of a trig function for 0, 30, 45, 60, or 90
degree reference angles
Reminder of how to use your hand to find the
value of a trig function for 0, pi sixths, pi
fourths, pi thirds and pi halves reference
angles.
Finding Exact Values
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Angles in Radians:
1. Determine reference angle
2. Change fraction to mixed numeral
3. Determine quadrant
4. Determine value using hand
5. Determine whether value is positive or
negative in that quadrant (All Scientists Take
Calculus)
Finding Exact Values - Degrees
If Angle is in degrees we will need to determine
our reference angle first by using the
following rules:
If the angle is in the first quadrant – it is a
reference angle
If the angle is in the second quadrant – subtract
the angle from 180.
Finding Exact Values - Degrees
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If the angle is in the third quadrant – subtract
180 from the angle
If the angle is in the fourth quadrant –
subtract the angle from 360
Finding Exact Values - Degrees
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(1) Determine whether value is positive or
negative from the quadrant
(2) Find reference angle – using preceding
rules
(3) Determine value of function using hand
Using Calculator to Approximate Value
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If angle is not one that uses one of the given
reference angles, calculator will be used to
approximate the value.
This value is not exact as the previous values
have been
Be careful that calculator is in correct
mode.
Using a Circle of Radius R
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To find the trig values given a point NOT ON
THE UNIT CIRCLE
Be sure to read the directions before finding
the six trig functions.
Six Trig Functions
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Tutorials
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More Tutorials
Applications – Projectile Motion
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The path of a projectile fired at an inclination θ
to the horizontal with initial speed v0 is a
parabola. The range of the projectile, that is the
horizontal distance that the projectile travels, is
found by using the formula
v02 sin(2 )
R
g
where g  32.2 ft / s 2 or 9.9m / s 2
Applications – Projectile Motion
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The projectile is fired at an angle of 45
degrees to the horizontal with an initial speed
of 100 feet per second. Find the range of the
projectile