section 4.4. in first section, we calculated trig functions for acute angles. in this section, we...
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Trig Functions of Any Angle
Section 4.4
In first section, we calculated trig functions for acute angles.
In this section, we are going to extend these basic definitions to cover any angle.
θ
θ
Plot the point (-3,4)
Label the hypotenuse r and find its length.
r43
22 r = 5
5
-3
4θ Sin θ =
Cos θ =
Tan θ =
54
53
34
Definitions of Trig Functions of Any AngleLet θ be an angle in standard position with
(x,y) a point on the terminal side. Then:
ry
rx
xy
Csc θ =
Sec θ =
Cot θ =
Sin θ =
Cos θ =
Tan θ =
yr
xr
yx
Find the 6 trig functions of θ given that the ray ends at the point (-15, -8)
-15
-8 17
178
1715
158
Csc θ =
Sec θ =
Cot θ =
Sin θ =
Cos θ =
Tan θ =
817
1517
815
Find the 6 trig functions of θ given that the ray ends at the point (12, -5)
12
-5 13
135
1312
125
Csc θ =
Sec θ =
Cot θ =
Sin θ =
Cos θ =
Tan θ =
513
1213
512
QuadrantsIn which quadrants was the Sine positive?
I and II
In which quadrants was the Cosine positive?I and IV
In which quadrants was the Tangent positive?I and III
Quadrants
All Trig Functionsare positive
Sine is positive
Cosine is positive
Tangent is positive
AllStudents
Take Calculus
What quadrant is θ in if:
a) Sin θ > 0 and Cos θ < 0
b) Tan θ > 0 and Cos θ < 0
c) Sin θ < 0 and Tan θ < 0
d) Cos θ > 0 and Tan θ > 0
→ II
→ III
→ IV
→ I
Given that Tan θ = - and Sin θ > 0, find the
remaining 5 trig functions of θ.24
7
What quadrant? II
-24
7 2525
7
2524
247
Csc θ =
Sec θ =
Cot θ =
Sin θ =
Cos θ =
Tan θ =
725
2425
724
Given that Cos θ = - and Sin θ < 0, find the
remaining 5 trig functions of θ.5
4
What quadrant? III
-4
5-3
53
54
43
Csc θ =
Sec θ =
Cot θ =
Sin θ =
Cos θ =
Tan θ =
35
45
34
Given that Sin θ = - and Tan θ < 0, find the
remaining 5 trig functions of θ.17
15
What quadrant? IV
-1517
817
15
178
815
Csc θ =
Sec θ =
Cot θ =
Sin θ =
Cos θ =
Tan θ =
1517
817
158
What did we learnHow to find the trig functions of an angle
given a point on its terminal side
How to determine the quadrant of an angle based on trig functions
How to find the trig functions based on one function and criteria
Homework: Page 297, 1-24 odd
Find the Sin, Cos, and Tan trig functions of θ given that the ray ends at the point (5,0)
5
y = 005
0
155
050
Sin θ =
Cos θ =
Tan θ =
Quadrant AnglesOn our Cartesian plane, we
have 5 critical points:
2
3
2
0
2
Find the Sine of these 5 angles
Sin 0 = 0
Sin = 1 2
Sin π = 0
Sin = -1 2
3
Sin 2π = 0
Graph of the Sine CurveUsing these 5 points, we can create the Sine
Curve
20
2
2
3
Quadrant AnglesUsing the same process, find the Cos of the 5
critical points.
Cos 0 = 1
Cos = 0 2
Cos π = -1
Cos = 0 2
3
Cos 2π = 1
Graph of the Cosine CurveUsing these 5 points, we can create the Sine
Curve
20
2
2
3
Reference AnglesThe acute angle formed by the terminal side
of an angle and the horizontal axis.
For an angle θ, we use θ’ to denote the reference angle
Reference AnglesWhat is the reference angle for
210º
Where is there an acute angle
between the terminal side of the
angle and the horizontal axis?
θ’ = 210 – 180 = 30º
Reference AnglesFind the reference angles for the following:
a) 330º
b) 225º
c) -225º
d) 750º
= 360º - 330º = 30º
= 225º - 180º = 45º
= -180º - -225º = 45º
= 750º - 720º = 30º
Reference AnglesIn general, for any angle θ
θ’ = θθ’ = 180 - θθ’ = π - θ
θ’ = θ - 180θ’ = θ - π
θ’ = 360 - θθ’ = 2π - θ
Reference AnglesFind the reference angle for
2nd Quadrant: → π – θ
= π –
=
4
3
4
3
4
Reference AnglesSo far, all we have been finding are reference
angles.
We use reference angles to find the exact value of angles that are not acute.
We will use this for the remainder of the year.
“GTK” – Good to Know
Finding the Exact Value1. Find the reference angle
2. Find the trig function of the reference angle
3. Check the sign of the function
Sin 200º1. Find the reference angle
2. Find the Sin of the reference angle
3. Is it positive or negative?
Cos 330º1. Find the reference angle
2. Find the Sin of the reference angle
3. Is it positive or negative?
Find the Sin, Cos, and Tan of 135ºReference Angle =
Quadrant =
Sin 135º =
Cos 135º =
Tan 135º =
Find the Sin, Cos, and Tan of -240ºReference Angle =
Quadrant =
Sin -240º =
Cos -240º =
Tan -240º =
Find the Sin, Cos, and Tan ofReference Angle =
Quadrant =
Sin =
Cos =
Tan =
4
7
4
7
4
7
4
7
Find the:a) Sin
b) Csc
c) Tan
d) Csc
e) Cot
4
5
3
2
6
11
6
7
3
4