what has to be broken before it can be used?. 5.3 and 5.4 understanding angles greater than...
TRANSCRIPT
What has to be broken before it can be used?
5.3 and 5.4 Understanding Angles
Greater than
We're going to explore how triangles in a Cartesian plane have trig ratios that relate
to each other
90
x
y
θ
Terminal Arm
Initial Arm
Angles, Angles, Angles
• An angle is formed when a ray is rotated about a fixed point called the vertex
Initial Arm (does not move)
Terminal Arm (the part that is rotated)
θ
Vertex
Think of Hollywood
Terminal Arm
Initial Arm
Depending on how hard the director wants
to snap the device, he/she will vary the
angle between the initial arm
and the terminal arm
• The trigonometric ratios have been defined in terms of sides and acute angles of right triangles.
• Trigonometric ratios can also be defined for angles in standard position on a coordinate grid.
x
y
Coordinate grid
Standard Position• An angle is in standard position if the vertex of
the angle is at the origin and the initial arm lies along the positive x-axis. The terminal arm can be anywhere on the arc of rotation
Greek Letters such as α,β,γ,δ,θ (alpha, beta, gamma, delta,
theta) are often used to define
angles!x
y
θ
Terminal Arm
Initial Arm
For example……
Initial arm
Not Standard Form Terminal Arm
Terminal Arm
Standard Form
Initial arm
Initial arm
Terminal Arm
Positive and Negative angles
A positive angle is formed by a
counterclockwise rotation of the terminal arm
Positive angles
θ
θ
A negative angle is formed by a clockwise rotation of
the terminal arm
Negative angles
The Four Quadrants
Quadrant IQuadrant II
Quadrant III Quadrant IV
The x-y plane is divided into four quadrants. If angle θ is a positive
angle, then the terminal arm lies in which
quadrant?
0º< θ < 90º
270 º < θ < 360º
90º < θ < 180º
180º < θ < 270º
Principal Angle and Related Acute Angle
The principal angle is the angle between the initial
arm and the terminal arm of an angle in standard position. Its angle is
between 0º and 360º
The principal angle is the angle between the initial
arm and the terminal arm of an angle in standard position. Its angle is
between 0º and 360º
The related acute angle is the acute angle between the terminal
arm of an angle in standard position
(when in quadrants 2, 3, or 4). and the x-axis. The related acute angle
is always positive and is between 0º and 90º
The related acute angle is the acute angle between the terminal
arm of an angle in standard position
(when in quadrants 2, 3, or 4). and the x-axis. The related acute angle
is always positive and is between 0º and 90º
Let’s look at a few examples……
θ
Terminal Arm
Initial Arm
βPrincipal Angle
Related Acute Angle
In these examples, θ represents the principal angle and β represents the
related acute angle
θ
Principal angle: 65º
θβ
Principal angle: 140 º
Related acute angle: 40º
θ
β
Principal angle: 225º
Related acute angle: 45º
θ
β Principal angle: 320º
Related acute angle: 40º
No related acute angle because the
principal angle is in quadrant 1
Notice anything?
•In the first quadrant the principal angle and related acute angle are always the same
•In the second quadrant we get the principal angle by taking (180º - related acute angle)
•In the third quadrant we can get the principal angle by taking (180º + related acute angle)
•In the fourth quadrant we can get the principal angle by taking (360º - related acute angle)
Let’s work with some numbers!
Angles Quadrant Sine
RatioCosineRatio
TangentRatio
Principal angle60º
1 0.8660 0.5 1.7320
Related acute angle (none)
Principal angle135º
2 0.7071 -0.7071 -1
Related acute
angle 45º
0.7071 0.7071 1
Principal angle220º
3 -0.6427 -0.7760 0.8391
Related acute
angle 40º0.6427 0.7760 0.8391
Principal angle300º
-0.8660 0.5 -1.7320
Related acute
angle 60º0.8660 0.5 1.7320
4
θ
SinSinθθ is positive is positive
CosCosθθ is positive is positive
TanTanθθ is positive is positive
Quadrant 1
θ
SinSinθθ = sin (180° - = sin (180° - θθ))
(180° - (180° - θθ))
-Cos-Cosθθ = cos (180° - = cos (180° - θθ))
-Tan-Tanθθ = tan (180° - = tan (180° - θθ))
Quadrant 2
θ
-Sin-Sinθθ = sin (180° + = sin (180° + θθ))
(180° + (180° + θθ))
-Cos-Cosθθ = cos (180° + = cos (180° + θθ))
TanTanθθ = tan (180° + = tan (180° + θθ))
Quadrant 3
θ
-Sin-Sinθθ = sin (360° - = sin (360° - θθ))
(360° - (360° - θθ))
CosCosθθ = cos (360° - = cos (360° - θθ))
-Tan-Tanθθ = tan (360° - = tan (360° - θθ))
Quadrant 4
AllAll ratios are positive ratios are positive
Summary
Only Only sinesine is positive is positive
Only Only cosinecosine is positive is positiveOnly Only tangenttangent is positive is positive
AS
T C
For any principal angle greater than 90 , the values of the primary trig ratios are either the same as, or the negatives of, the ratios for the related acute angle
When solving for angles greater than 90 , the related acute angle is used to find the related trigonometric ratio. The CAST rule is used to determine the sign of the ratio
Quadrant IQuadrant II
Quadrant III Quadrant IV
Cosine
AllSine
Tangent
CAST Rule
Example1.
Point P(-3,4) is on the terminal arm of an angle in standard position.
a)Sketch the principal angle θ
b) Determine the value of the related acute angle to the nearest degree
c) What is the measure of θ to the nearest degree?
Solutiona) Point P(-3,4) is in quadrant 2, so the principal angle θ terminates in quadrant 2.
b) The related acute angle β can be used as part of a right triangle with sides of 3 and 4. We can figure out β using SOHCAHTOA.
P(-3,4)
θ
-3
4
β
127
53180
180
Note…..Whenever we make a triangle such as the one above there is something important to remember…THE HYPOTENUSE will always be expressed as a positive value, regardless of the quadrant in which it
occurs!! Lets look at an example….
53
54
sin
54
1
hyp
oppSin
Example 2
Point (3,-4) is on the terminal arm of an angle in standard position
a) What are the values of the primary trigonometric functions?
b) What is the measure of the principal angle θ to the nearest degree?
SolutionAssuming that you can draw a circle around the x-y axis, with your point lying somewhere on
the perimeter, then it would follow that the hypotenuse of our right angled triangle would be the same as the radius of the circle.
Using pythagorean theorem, we find that r = 5 (note it is positive regardless of the quadrant. Using these values, then
3
-4 r =5
θ
5
4
Bsin
r
y
5
3
cos
r
xB
3
4
tan
x
yB
To evaluate B, select cosine and solve for B. Using cos B gives us
53
5
3cos 1
B
B
……….
From the sketch, clearly θ is not 53°. This angle is the related acute angle. In this case θ = 360°-53° = 307°
Just as a side note….once again notice that if you take the cos of 307° you get 0.6018 and if you take the cos of 53° you also get 0.6018
Well Ross, what is
it?
Wait for it, wait for it…
Ok, hmk
WOOOOW!