Unit 8 Trigonometric Functions Radian and degree measure Unit Circle Right Triangles Trigonometric...
53
Unit 8 Trigonometric Functions • Radian and degree measure • Unit Circle • Right Triangles • Trigonometric functions • Graphs of sine and cosine • Graphs of other Trigonometric functions • Inverse Trigonometric functions • Applications and Models
Unit 8 Trigonometric Functions Radian and degree measure Unit Circle Right Triangles Trigonometric functions Graphs of sine and cosine Graphs of other
Unit 8 Trigonometric Functions Radian and degree measure Unit
Circle Right Triangles Trigonometric functions Graphs of sine and
cosine Graphs of other Trigonometric functions Inverse
Trigonometric functions Applications and Models
Slide 2
13.1 Right Triangle Trigonometry Evaluate trigonometric
functions of acute angles Use the fundamental trigonometric
identities Use trigonometric functions to model applied
problems
Slide 3
Opp. hypotenuse Remember the trigonometric functions used to
solve right triangles? Adjacent
Slide 4
In addition to the trigonometric functions we have already
learned, we have three more. Notice they are merely the reciprocals
of the three we already have.
Slide 5
A B C 4 3 5 Find the six trig functions of angle A:
Slide 6
It is also helpful to remember the special right triangle
relationships Now with all of this information, we are able to find
trigonometric values of many angles on the unit circle.
Slide 7
Inverse Sine Function sin -1 x or arcsin x Inverse Cosine
Function cos -1 x or arcos x tan-1 x or arctan x Inverse Tangent
Function Inverse trigonometric functions help us to find the angle
when we are given the triangle ratios What this question is asking
is, what is the angle that has a cosine of ?
Slide 8
x 5 25 Solve for x. Sketch a triangle that has an acute angle ,
and then find . 5 8
Slide 9
Slide 10
If you look up from the middle of the football field to the top
of the gym, you see the angle of elevation is 41 degrees. If you
are 250 feet from the base of the gym, how tall is the gym? 41 250
feet x 1) Draw picture first! 2) Set up trig ratio 3) Solve
Slide 11
5150 x y An airplane is flying at an elevation of 5150 ft,
directly above a straight highway. Two motorists are driving cars
on the highway on opposite sides of the plane, and the angle of
depression to one car is 35 and to the other is 52 . How far apart
are the cars? 11378.58 feet 52 35 52 35
Slide 12
13.2 Radian and Degree Measure Describe angles Use radian
measure Use degree measure Use angles to model applied
problems
Slide 13
Positive angle Negative angle Terminal side of angle Initial
side of angle This is known as standard position
Slide 14
Alpha and Beta have the same initial and terminal sides. Angles
like this are called coterminal. To find positive or negative
coterminal angles: Add or Subtract 360 or 2 . Coterminal Angles: 2
angles who have the same terminal sides.
Slide 15
A radian is the measure of the angle that intercepts an arc
whose length is equal to the radius of the circle.
Slide 16
A quick conversion guide: Reminder- supplementary angles are
two angles that add up to 180 degrees or radians. Complimentary
angles are two angles that add up to 90 degrees or radians
Slide 17
Find the radian measure of the angle: Find the degree measure
of the angle: a)b) Find 1 positive & 1 negative angle that are
coterminal with the angle: a) b) Are the angles coterminal? no
Slide 18
Arc Length In radians s r Find s if the angle is 2 radians and
the radius is 5cm s = (5)(2) = 10 cm Find the length of an arc that
subtends a central angle of 45 in a circle of radius 10 m.
Slide 19
A sector of a circle of radius 24 mi has an area of 288 mi 2.
Find the central angle of the sector. Area of a Circular Sector: A
= sector area r = radius = angle measure (rad) r
Slide 20
Slide 21
13.6 Trigonometric functions: The Unit Circle Identify a unit
circle and its relationship to real numbers Evaluate trigonometric
functions using the unit circle Use the domain and period to
evaluate sine and cosine functions Use a calculator to evaluate
trigonometric functions
Slide 22
OP = 1 OR = x RP = y o p (x, y) R (1,0) (0,-1) (-1,0)
(0,1)
Slide 23
O (1,0) (0,-1) (-1,0) (0,1) This means the point P at (x, y)
can be represented by the ordered pair (cos x, sin x). It is easy
for us to figure out the trigonometric functions if the terminal
side of an angle falls on an axis.
Slide 24
It is also helpful to remember the special right triangle
relationships Now with all of this information, we are able to find
trigonometric values of many angles on the unit circle.
Slide 25
Slide 26
What is the sine, cosine, tangent of 150 degrees? A reference
angle is the acute angle formed between the terminal side and the
x-axis
Slide 27
Cosine also has a period of Tangent has a period of The sine
function repeats itself every radians (or 360 ). The smallest
number in which the function repeats itself is called the period.
So, the sine function has a period of
Slide 28
13.3 Trigonometric functions of any angle Evaluate
trigonometric functions of any angle Use reference angles to
evaluate trigonometric functions Evaluate trigonometric functions
of real numbers
Slide 29
o p (x, y) OP = r Let us re-visit the circle of radius r. We
can still find the length of x and y for any angle using trig. x y
What is the length of y? What is the length of x?
Slide 30
This information allows us to find the sine and cosine for any
angle.
Slide 31
o ALL I IV II III The positive trig. functions are listed.
tangent cosine sine All-all Students-sine Take-tangent
Calculus-cosine Use the reference angle formed between the terminal
side and the x-axis
Slide 32
Find the reference angle for the given angle:
Slide 33
1.Draw a picture 2.Find (ref angle) 3.Trig ( ) = Trig ( )
4.Decide if its POS or NEG Evaluating Trig Functions: Find the
value of the other 5 trig functions of given the following: We know
it is in 4 th quadrant 5 1
Slide 34
1 (1, 0) (0, 1) (0, -1) (-1, 0)
Slide 35
Slide 36
13.7/13.8 Graphs of trig functions Sketch the graphs of basic
sine and cosine functions Use amplitude and period to help sketch
the graphs of sine and cosine functions Sketch the translations of
the graphs of sine and cosine functions Use the sine and cosine
functions to model applied problems
Slide 37
1 (1, 0) (0, 1) (0, -1) (-1, 0)
Slide 38
1 (1, 0) (0, 1) (0, -1) (-1, 0)
Slide 39
1
Slide 40
1
Slide 41
When the angle is multiplied, the period is changed. 1
Slide 42
When the angle is added to or subtracted from a number, the
graph is shifted to the left or right. This is called a phase shift
1
Slide 43
When the entire functions is added to or subtracted from a
number, the graph is shifted up or down. This is called a vertical
translation 1
Slide 44
Review for Sine & Cosine Functions period = 2 for f(x) =
sin x & f(x) = cos x * Domain = ( , ) Range = [1, 1] y =
asin(k(x b))+ c OR y = acos(k(x b))+ c Amplitude (half the distance
between the max and the minimum of the function. Or the distance
above or below the midline) = |a| Period = Phase Shift (Lt or Rt) =
b Graphing Interval = Range: Vertical Shift (Up or Dn)= c Factoring
may be necessary to get it in this form!!!
Slide 45
Amplitude = 3 1 Set graphing interval first Sketch the graph.
State the amplitude & period. Divide period into four parts and
plot the points accordingly
Slide 46
Sketch the graph. State the amplitude & period. Set
graphing interval first 1 Amplitude = 1 Divide period into four
parts and plot the points accordingly
Slide 47
y = a tan k (x b) y = a cot k (x b) Tangent & Cotangent
Functions Period = Phase Shift (Lt or Rt) = b Graphing Interval =
[b, b + ] Tangent has a zero at the beginning, zero at the end, and
an asymptote in the middle. Cotangent has an asymptote at the
beginning and end, and a zero in the middle.
Slide 48
Tangent has a zero at the beginning, zero at the end, and an
asymptote in the middle.
Slide 49
Cotangent has an asymptote at the beginning and end, and a zero
in the middle.
Slide 50
y = a sec k (x b) + c y = a csc k (x b) + c Secant &
Cosecant Functions Amplitude (vertical distance between Us and
midline) = Period = Phase Shift (Lt or Rt) = b Graphing Interval =
[b, b + ] C is the midline
Slide 51
1 Since it is the reciprocal, where the sine function was equal
to zero there is now an asymptote for cosecant function at
0,180,360 degrees.
Slide 52
1 Similarly, where the cosine function was equal to zero there
is now an asymptote for the secant function at 90,270 degrees.
Slide 53
Key Chapter points: Radian and degree measure Unit Circle Right
Triangles Trigonometric functions Graphs of sine and cosine Graphs
of other Trigonometric functions Inverse Trigonometric
functions