Download Trigonometric Function Graphs

Trigonometric Function Graphs
General Right Triangle
General Trigonometric Ratios
SOH CAH TOA
B
c
a
A
b
C
Pythagorean Theorem, 𝑐 2 = 𝑎2 + 𝑏2
𝑂𝑃𝑃
𝑠𝑖𝑛 𝜃 =
𝐻𝑌𝑃
𝐴𝐷𝐽
𝑐𝑜𝑠 𝜃 =
𝐻𝑌𝑃
𝑂𝑃𝑃
𝑡𝑎𝑛 𝜃 =
𝐴𝐷𝐽
𝑐𝑠𝑐 𝜃 =
𝐻𝑌𝑃
𝑂𝑃𝑃
𝐻𝑌𝑃
𝑠𝑒𝑐 𝜃 =
𝐴𝐷𝐽
𝑐𝑜𝑡 𝜃 =
𝐴𝐷𝐽
𝑂𝑃𝑃
𝑂𝑃𝑃
𝑆𝑖𝑛 𝜃 =
𝐻𝑌𝑃
𝐴𝐷𝐽
𝐶𝑜𝑠 𝜃 =
𝐻𝑌𝑃
𝑂𝑃𝑃
𝑇𝑎𝑛 𝜃 =
𝐴𝐷𝐽
Solutions for Non-Right Triangles
Law of Cosines
Law of Sines
Two Examples of Non-Right
Triangles
B
c
c
A
B
a
a
b
C
A
b
C
Solutions for Non-Right Triangles
Law of Cosines
Law of Sines
𝑎2 = 𝑏2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
𝑏2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 cos 𝐵
sin 𝐴
sin 𝐵
sin 𝐶
=
=
𝑎
𝑏
𝑐
𝑐 2 = 𝑎2 + 𝑐 2 − 2𝑎𝑏 cos 𝐶
B
or
c
𝑎
𝑏
𝑐
=
=
sin 𝐴
sin 𝐵
sin 𝐶
A
a
𝑏2 + 𝑐 2 − 𝑎2
2𝑏𝑐
𝐵 = 𝑐𝑜𝑠 −1
𝑎2 + 𝑐 2 − 𝑏2
2𝑎𝑐
𝐶 = 𝑐𝑜𝑠 −1
𝑎2 + 𝑏2 − 𝑐 2
2𝑎𝑏
𝐴 = 𝑐𝑜𝑠
C
b
−1
B
c
A
a
b
C
These equations work no matter the type of triangle, acute, right, or obtuse.
The Unit Circle
Radius = 1 Unit Length
Sine Graph
- Parent function is an Odd Function, which means given an (x, y), (x, f(x)), on
the graph there has to be a (– x, – y). Stated another way, given an f(x) then for
ordered pairs (– x, – y) = (– x, f(– x) = (– x, – f(x))  sin(–) = – sin().
y
- Its period is 2.
- Since it is an Odd Function, it is symmetric about the origin. The Cosecant,
the reciprocal of the Sine Function is also an Odd Function.
1
- Domain - all Reals
- Range [–1, 1]
0.5
- It is positive in the First and Second Quadrant, and negative in the Third and
Fourth Quadrant.
x
-π/2
-π/4
π/4
-0.5
-1
π/2
3π/4
π
5π/4
3π/2
7π/4
2π
Cosecant Graph
y
- Parent function is an Odd function, which means given an (x, y), (x, f(x)), on the
graph there has to be a (– x, – y). Stated another way, given an f(x) then for
ordered pairs (– x, – y) = (– x, f(– x) = (– x, – f(x))  csc(–) = – csc().
𝒈 𝒙 = 𝒄𝒔𝒄 𝜽
- Its period is 2.
3
- Since it is an Odd Function, it is symmetric about the origin.
2.5
- Domain - all Reals except multiples of 𝜋 + 𝑛𝜋, 𝑤ℎ𝑒𝑟𝑒 𝑛 = ±0, ±1, ±2, …
2
- Range [– , –1]  [1, ]
1.5
- It is positive in the First and Second Quadrant, and negative in the Third and
Fourth Quadrant.
1
0.5
𝒇 𝒙 = 𝒔𝒊𝒏 𝛉
x
-π/2
-π/4
π/4
-0.5
-1
-1.5
-2
-2.5
-3
π/2
3π/4
π
5π/4
3π/2
7π/4
2π
Cosine Graph
- Parent function is an Even Function, which means given an (x,
y), (x, f(x)), on the graph there has to be a (– x, y). Stated
another way, given an f(x) then for ordered pairs (– x, y) = (– x,
f(– x) = (– x, f(x))  cos(– ) = cos()
y
- Its period is 2.
- Since it is an Even Function, it is symmetric about the y-axis.
The Secant, the reciprocal of the Cosine Function, is also an
Even Function.
1
- Domain - all Reals
- Range [–1, 1]
0.5
- It is positive in the First and Fourth Quadrant, and negative in
the Second and Third Quadrant
x
-π/2
-π/4
π/4
-0.5
-1
π/2
3π/4
π
5π/4
3π/2
7π/4
2π
Secant Graph
- Parent function is an Even Function, which means given an
(x, y), (x, f(x)), on the graph there has to be a (– x, – y).
Stated another way, given an f(x) then for ordered pairs (– x,
– y) = (–x, f(– x) = (– x, – f(x))  sec(– ) = sec()
y
𝒈 𝒙 = 𝒔𝒆𝒄 𝜽
- Its period is 2.
3
- Since it is an Even Function, it is symmetric about the yaxis.
2.5
𝜋
2
- Domain is all Reals except 2 + 𝑛𝜋, 𝑛 = 0, 1, 2, …
1.5
- Range [– , –1]  [1, ]
1
𝒇 𝒙 = 𝒄𝒐𝒔 𝛉
- It is positive in the First and Fourth Quadrant, and negative
in the Second and Third Quadrant.
0.5
x
-π/2
-π/4
π/4
-0.5
-1
-1.5
-2
-2.5
-3
π/2
3π/4
π
5π/4
3π/2
7π/4
2π
Tangent Graph
- Parent function is an Odd Function, which means given an (x, y), (x, f(x)), on
the graph there has to be a (– x, – y). Stated another way, given an f(x) then for
ordered pairs (– x, – y) = (– x, f(– x) = (– x, – f(x))  tan(– ) = – tan().
- Its period is .
y
3
- Since it is an Odd Function, it is symmetric about the origin.
𝜋 𝜋
- Domain − 2 , 2 , 𝑤ℎ𝑖𝑐ℎ 𝑚𝑒𝑎𝑛𝑠 𝑖𝑡 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑖𝑛𝑐𝑙𝑢𝑑𝑒
± 1, ±2, …
𝜋
2
2.5
+ 𝑛 − 1 𝜋, 𝑛 =
2
1.5
- Range, all Reals
1
- It is positive in the First and Third Quadrant, and negative in the Second and
Fourth Quadrant.
0.5
x
-π/2
-π/4
π/4
-0.5
-1
-1.5
-2
-2.5
-3
π/2
Cotangent Graph
- Parent function is an Odd Function, which means given an (x, y), (x, f(x)), on
the graph there has to be a (– x, – y). Stated another way, given an f(x) then for
ordered pairs (– x, – y) = (– x, f(– x) = (– x, – f(x))  cot(– ) = – cot().
y
- Its period is .
3
- Since it is an Odd Function, it is symmetric about the origin.
2.5
- Domain all Reals not to include multiples of 
2
- Range All Reals
1.5
1
- It is positive in the First and Third Quadrant, and negative in the Second and
Fourth Quadrant.
0.5
x
-π
-3π/4
-π/2
-π/4
π/4
-0.5
-1
-1.5
-2
-2.5
-3
π/2
3π/4
π
Arc Functions
Arcsine, sin–1, Function
- It is the inverse function of the sine, which means sin–1(sin ) =  and sin(sin–1 ½) = ½.
- Domain [–1, 1]
y
𝜋 𝜋
- Range − 2 , 2
- The value of the arcsine is an angle.
π/2
π/4
x
-1
-0.5
0.5
-π/4
-π/2
1
Arccosine, cos–1, Function
- It is the inverse function of the cosine, which means cos–1(cos ) =  and cos(cos–1 ½) = ½.
- Domain [–1, 1]
y
- Range [0, ]
- The value of the arccosine is an angle.
π
3π/4
π/2
π/4
x
-1
-0.5
0.5
1
Arctangent, tan–1, Function
- It is the inverse function of the tangent, which means tan–1(tan ) =  and tan(tan–1 ½) = ½.
y
3π/4
- Domain is all Reals
𝜋 𝜋
- Range − 2 , 2 ; NOTE it does not include the end points of the interval.
- The value of the arctangent is an angle.
π/2
π/4
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
-π/4
-π/2
2
3
4
5
6
7
8
9
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