reference angles and trigonometry using trigonometry in a right triangle we were limited to acute...
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Reference AnglesAnd Trigonometry
Using Trigonometry in a Right TriangleWe were limited to Acute Angles
We can extend Trigonometry to Angles of Any Measureby placing those angles in the coordinate plane
We do this by using reference angles,Acute Angles measured to the x-axis.
Angles are Placed with one sidecalled the initial side on the positive x-axis.
The terminal side is rotated counter-clockwise.
135°
A Reference Angle is measuredto the x-axis.
The terminal side is rotated counter-clockwise.
135°
45°
A Reference Angle is measuredto the x-axis.
The terminal side is rotated counter-clockwise.
225°
45°
A Reference Angle is measuredto the x-axis.
The terminal side is rotated counter-clockwise.
315°
45°
A Reference Angle is measuredto the x-axis.
If the terminal side is rotated clockwise, the angle measure isNegative.
-45°
45°
Always Positive.
, 180 0, 02, 360
3
2
2
1,0
0,1
-1,0
0, -1
Unit Circle has a radius
of 1 unit.
, 180 0, 02, 360
3
2
2
1,0
0,1
-1,0
0, -1
Unit Circle has a radius
of 1 unit.
45°
45°
1
2
2
2
2
Cos
+
Sin
+
2
2
2
2
x=
=y
Cosine = xSine = y
, 180 0, 02, 360
3
2
2
1,0
0,1
-1,0
0, -1
45°45°
Cos
+
Sin
+
135°
Reference Angle =
Cos
-
Sin
+
, 180 0, 02, 360
3
2
2
1,0
0,1
-1,0
0, -1
45°
45°
Cos
+
Sin
+
225°
Reference Angle =
Cos
-
Sin
+
Cos
-
Sin
-45°
135°45°
225°
, 180 0, 02, 360
3
2
2
1,0
0,1
-1,0
0, -1
45°
45°
Cos
+
Sin
+
315°
Reference Angle
Cos
-
Sin
+
Cos
-
Sin
-45°
135°45°
225°
45°
Cos
+
Sin
-
315°
, 180 0, 02, 360
3
2
2
1,0
0,1
-1,0
0, -1
45°
45°
Cos
+
Sin
+Cos
-
Sin
+
Cos
-
Sin
-45°
135°45°
225°
45°
Cos
+
Sin
-
315°
Quadrant 2Quadrant 1
Quadrant 4Quadrant 3
, 180 0, 02, 360
3
2
2
1,0
0,1
45°
Cos
+
Sin
+
45°
Quadrant 1
Cosine = xSine = y
Tangent = Δy Δx
Tangent = Sine Cosine
1
22
22
tan
tan = 1
, 180 0, 02, 360
3
2
2
1,0
0,1
-1,0
0, -1
45°
45°
Cos
+
Sin
+Cos
-
Sin
+
Cos
-
Sin
-45°
135°45°
225°
45°
Cos
+
Sin
-
315°
Quadrant 2Quadrant 1
Quadrant 4Quadrant 3
tan = 1tan = -1
tan = -1tan = 1
, 180 0, 02, 360
3
2
2
1,0
0,1
-1,0
0, -1
Cos
+
Sin
+Cos
-
Sin
+
Cos
-
Sin
-
Cos
+
Sin
-
Quadrant 2Quadrant 1
Quadrant 4Quadrant 3
Tan
-
Tan
-T
an +
Tan
+
Tangent = Sine Cosine
, 180 0, 02, 360
3
2
2
1,0
0,1
-1,0
0, -1
30°
Cos
+
Sin
+
30°
1
2
1
2
3
2
1
2
3
Cosine = xSine = y
, 180 0, 02, 360
3
2
2
1,0
0,1
-1,0
0, -1
30°30°
Cos
+
Sin
+Cos
-
Sin
+
Cos
-
Sin
-30°
150°30°
210°
30°
Cos
+
Sin
-
330°
150°
, 180 0, 02, 360
3
2
2
Cos
+
Sin
+Cos
-
Sin
+
Cos
-
Sin
-
150°30°
210°
Cos
+
Sin
-
330°
3
3
3
1
3
2
2
1
23
21
)30tan(
3
3)30tan( 3
3)150tan(
3
3)270tan(
3
3)330tan(
Tangent = Sine Cosine
, 180 0, 02, 360
3
2
2
1,0
0,1
-1,0
0, -1
60°
Cos
+
Sin
+
60°
1
2
1
2
32
1
2
3
, 180 0, 02, 360
3
2
2
1,0
0,1
-1,0
0, -1
60°
Cos
+
Sin
+
60°
2
1
2
3
60°
120°
Cos
+
Sin
+
120°
, 180 0, 02, 360
3
2
2
1,0
0,1
-1,0
0, -1
60°
Cos
+
Sin
+
60°
2
1
2
3
60°
Cos
-
Sin
+
120°
60°
Cos
-
Sin
-
240°
, 180 0, 02, 360
3
2
2
1,0
0,1
-1,0
0, -1
60°
Cos
+
Sin
+
60°
60°
Cos
-
Sin
+
120°
60°
Cos
-
Sin
-
240°
60°
Cos
+
Sin
-300°
, 180 0, 02, 360
3
2
2
1,0
0,1
-1,0
0, -1 C
os +
Sin
+
60°
Cos
-
Sin
+
120°
Cos
-
Sin
-
240°C
os +
Sin
-300°
31
2
2
3
21
23
)60tan(
Tangent = Sine Cosine
3)240tan(
3)120tan( 3)60tan(
3)300tan(
, 180 0, 02, 360
3
2
2
Cosine = xSine = y
Cos
+
Sin
(1 , 0)(-1 , 0)
(0 , 1)
(0 , -1)
Cos(0) =1Sin(0) = 0
Cos
Sin
+ Cos(90) =0Sin(90) = 1
90°180°
Cos
-
Sin
Cos(180) = -1Sin(180) = 0
270°
Cos
Sin
- Cos(270) = 0
Sin(270) = -1
, 180 0, 02, 360
3
2
2
Cosine = xSine = y
Tangent = Sine Cosine
Cos
+
Sin
(1 , 0)(-1 , 0)
(0 , 1)
(0 , -1)
Cos(0) =1Sin(0) = 0
Cos
Sin
+ Cos(90) =0Sin(90) = 1
90°180°
Cos
-
Sin
Cos(180) = -1Sin(180) = 0
270°
Cos
Sin
- Cos(270) = 0
Sin(270) = -1
Tan(0) =0
Tan(90) undefined
Tan(180) =0
Tan(270) undefined
Evaluate the trigonometric functions at each real number.
2
3,
2
1
3
2Sin
3
2Cos
3
2Tan
= y
= x
x
y
2
3
2
1
2
1
2
3
1
2
2
3
3
1203
2
Evaluate the six trigonometric functions at each real number.
(0, -1)2
2Sin
2Cos
2Tan
= y
= x
= -1
= 0
x
y
0
1DNE
Does Not Exist
2Sec
0
1 DNE
2Cot
1
0
2Csc
= -1
= 0
Evaluate the six trigonometric functions at each real number.
4
7
2
2,
2
2
Sin
4
7
4
7Cos
4
7Tan
4
7Csc
4
7Sec
4
7Cot
2
2
2
2
-1-1
2
2
So, you think you got it now?