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13.1 -- Using Trig with Right Triangles
Wednesday, April 30, 2014
7:07 AM
SOH CAH TOA
sin θ =
csc θ =
cos θ =
sec θ =
tan θ =
cot θ =
Example 1:
Find the 6 trig functions of the angle θ.
5
12
Extra Ex1:
Find the 6 trig functions of angle A in triangle ABC with AC = 24 and CB = 7.
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Example 2:
If θ is an acute angle of a right triangle and
, what is tan θ?
If θ is an acute angle of a right triangle and
, what is of csc θ?
Evaluate the 6 trig functions of the angle
2.
1.
In a right triangle, is an acute and
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3.
. What is
Special Trig Values
Find the value of
for the right triangle shown.
8
30
Extra Ex3:
Find the value for x.
6
x
45⁰
Solving a triangle means that you find all of the angles and all of the sides.
Example 4:
Solve the triangle by using a calculator.
Round your answers to the nearest tenth.
A
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28
15
B
C
Extra Ex4:
Solve the triangle by using a calculator.
Round your answers to the nearest tenth.
A
54⁰
C
20
B
**They are always measured
from a horizontal line up or
from a horizontal line down!
A parasailer is attached to a boat with a rope 300 feet long. The angle of elevation
from the boat to the parasailer is 48 Estimate the parasailer's height above the boat.
Extra Ex6:From a point on the ground 28 ft from the base of a flagpole, the angle
of elevation to the top of the flagpole is 63⁰. Estimate the height of the flagpole.
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13.2 -- Define General Angles and Use Radian
Measures
Thursday, May 1, 2014
8:53 AM
Angles in Standard Position
Animation of terminal side of an angle
Example 1:
Draw an angle in standard position of the given measure.
1. 240⁰
2. 500⁰
3.
animation of coterminal angles
The angles
and
are
coterminal because their terminal sides
coincide. An angle coterminal with a given
angle can be found by adding or
subtracting multiples of
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50⁰
Find one positive angle and one negative angle that are coterminal with the given
angle
1.
2.
Draw an angle with the given measure in standard position. Then find one
positive coterminal angle and one negative coterminal angles.
1. 65⁰
2. 230⁰
3. 740⁰
Radian Measure
One radian is the measure of an angle in
standard position whose terminal side
intercepts an arc of length
Because the circumference of a circle is
,
there are
radians in a full circle. Degree
measure and radian measure are therefore
related by the equation
radians,
or
radians,
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Degrees to Radians
Radians to Degrees
Multiply by
Multiply by
1. Convert
to radians
2. Covert
radians to degrees.
Covert the degree measures to radians
and the radian measures to degrees.
1. 135⁰
2. - 50⁰
3.
4.
Arc Length:
Area:
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A softball field forms a sector with the dimensions
shown. Find the length of the outfield fence and
the area of the field.
Extra Ex4:
Find the arc lenth and the area of a sector that is formed by a 120⁰ angle and a
radius of 18 ft.
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13.3 -- Evaluate Trig Functions of Any Angle
Thursday, May 1, 2014
10:05 AM
General Definitions of Trig Functions
Let be an angle in standard position, and let
be the point where the terminal side of
intersects the circle
. The six
trigonometric functions or are defined as
follows:
These functions are sometimes
called circular functions
Ex1: Let
be a point on the terminal side
of an angle in standard position. Evaluate the
six trigonometric functions of .
Note: you can always use
Extra Ex1:
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Extra Ex1:
Let
be a point on the terminal side of an angle θ in the standard position.
Find each of the 6 trig functions of θ.
The Unit Circle
animation of Quadrantal Angles
The circle
, which has a center
and a radius of 1, is called the
unit circle. The values of
are the same as
** for the unit circle sin θ = y, cos θ = x, and tan θ =
Ex2: Use the unit circle to evaluate the
6 trig functions of
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(or the slope)
Extra Ex2:Find all six trig functions
when θ = 180⁰
Reference Angle = how far away the given
angle is away from the -axis.
animation of reference angles
THE UNIT CIRCLE
Green: 30 reference angles:
Red: 45 references angles:
Blue: 60 reference angles:
Ex3:
Find the reference angle:
2.
1.
Use reference angles to evaluate the given functions.
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4.
Use reference angles to evaluate the given functions.
2.
1.
Extra Ex4: Evaluate
1. sin(-225⁰)
2. cot
Guided practice for examples 3 and 4.
Sketch the angle, then find the reference angle.
1.
5. Evaluate
3.
2.
without a calculator.
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4.
The "frogbot" is a robot designed for exploring rough terrain on other planets. It can
jump at a 45° angle and with an initial speed of 16 feet per second. On Earth, the
horizontal distance (in feet) traveled by a projectile launched at an angle and with
and initial speed (in ft. per sec) is given by:
How far can the frogbot jump on Earth?
Rock Climbing: A rock climber is using a rock climbing treadmill that is 10.5 feet
long. The climber begins by lying horizontally on the treadmill, which is then
rotated about its midpoint by 110° so that the rock climber is climbing towards the
top.. If the midpoint of the treadmill is 6 feet above the ground, how high above
the ground is the top of the treadmill?
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13.4 -- Evaluate Inverse Trig Functions
Thursday, May 1, 2014
1:37 PM
Inverse Sine
If
then
Sine
or
Cosine
Inverse Cosine
If
then
or
Tangent
Inverse Tangent
If
then
or
Ex1: Evaluate inverse trig functions
Evaluate the expression in both radians and degrees.
1.
2.
Use the values from your unit circle.
3.
Extra Ex1: Evaluate the expressions in both radians and degrees.
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Extra Ex1: Evaluate the expressions in both radians and degrees.
1. sin-1 (-1)
2. cos-1
Ex2: Solve a Trig Equation
Solve the equation
where
3. tan-1
Guided Practice for Examples 1 and 2
Evaluate the expression in both radians and degrees.
2.
1.
Solve the equation for
5.
7.
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3.
4.
What is the measure of the angle in the triangle shown?
A) 28.6°
B) 33.1°
C) 56.9°
D) 61.4°
Ex4: Write and solve a trig equation
Monster Trucks: A monster truck drives off a ramp
in order to jump onto a row of cars. The ram has a
height of 8 ft. and a horizontal length of 20 ft.
What is the angle of the ramp?
Guided Practice for examples 3 and 4
Find the measure of the angle
1.
2.
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3.
13.5 -- Applying the Law of Sines
Thursday, May 1, 2014
2:12 PM
Law of Sines can be used to solve triangles in which the
following are given ASA or AAS
or SSA -- this is called the ambiguous case as there may be
0,1,or 2 triangles with those measures
Law of Sines
Ex1: Solve a triangle for the AAS or the ASA case
Solve
with
Extra Ex1: Solve
.
1. B = 34°, C = 100°, b = 8
2. A = 51°, B = 44°, c = 11
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If we are given SSA then 0, 1, or 2 triangles may exist.
Note: In the ambiguous case we are given 2 sides and an angle opposite one of those sides.
For ex: suppose b, c, and C are given. This situation is ambiguous because C is opposite c.
• No Triangles -- when you use the law of sines and the result is undefined.
• 1 triangle -- If the first angle you solve for is 90⁰, you have one right triangle
• 2 triangles -- if you solve for the first angle , then let
, and if
, then a 2nd triangle exists!
Ex2: Solve the SSA case with one solution
Solve
with
Ex3: Examine the SSA case with no solution.
Solve
with
Ex4: Solve the SSA case with two solutions
Solve
with
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Solve
1. A = 122°, a = 18, b = 12
2. A = 36°, a = 9, b = 12
3. A = 50°, a = 2.8, b = 4
4. B = 105°, b = 13, a = 6
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Finding the area of a triangle when you are given SAS.
** another way of saying this, is that the area is half the
product of 2 sides and the sine of the angle between them.
Find the Area of the triangular region.
A
Finding the Area of a Triangle (SAS)
Find the area of ΔABC.
34.3 ft
B
Find the area of ΔDEF
D
12.7 ft
F
127⁰
9.3 ft
E
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55.2⁰
42 ft
C
13.6 -- Applying the Law of Cosines
Friday, May 9, 2014
12:20 PM
If we are given SAS or SSS, then we must use Law of Cosines.
For ex: it would be impossible to create a triangle
with side lengths of 3, 4, and 10.
Ex1: Solve a triangle for the SAS case
Solve
with
Use the law of cosines to find the side length .
Ex2: Solve a triangle for the SSS case
Solve
with
Guided practice for examples 1 and 2
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Guided practice for examples 1 and 2
1. a = 8, c = 10, B = 48°
2. a = 14, b = 16, c = 9
Heron's Area Formula -- can only be used when you know SSS
1) Find the semi-perimeter (s) (half of the perimeter of a triangle)
2) Then use the following formula to find the area.
Ex4: Solve a multi-step problem
Urban Planning: The intersection of 3 streets
forms a piece of land called a traffic triangle.
Find the area of the traffic triangle shown.
Guided Practice for example 4
Find the area of
1.
2.
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3.