trigonometric ratios in the unit circle 14 april 2011
TRANSCRIPT
Trigonometric Ratios in the Unit Circle
14 April 2011
Trigonometric Ratios in the Unit Circle
The unit circle has a radius of 1
tcosr
xtcos
tsinr
ytsin
Trigonometric Ratios in the Unit Circle, cont.
tsecx
rtsec
tcscy
rtcsc
The tangent and cotangent formulas stay the same
“All Students Take Calculus”AS
CT
all ratios are positive
sine is positive
tangent is positive
cosine is positive
cosecant is positive
cotangent is positive
secant is positive
Example:
Trigonometric Ratio
Sine
Cosine
Tangent
Cosecant
Secant
Cotangent
5
Example:
Trigonometric Ratio
Sine
Cosine
Tangent
Cosecant
Secant
Cotangent
18
31
Your Turn:
On the Signs of Trigonometric Ratios handout, complete the feature map and problems 1 – 3
Graphing Negative Radians Find the positive
coterminal angle 1st! Sketch the positive
coterminal angle
6
11
6
112
6
1
6
Graphing Negative Radians, cont.
3
3
12
3
53
5
Graphing Radians with Multiple Revolutions If the angle measure is
larger than 2 pi, keep subtracting 2 from the fraction until the fraction is between 0 and 2 pi. (Find a coterminal angle between 0 and 2 pi.)
3
2
3
22
3
8
3
10
Graphing Multiple Rev. Radians, cont.
5
35
32
5
135
132
5
235
23
Your Turn:
On the Signs of Trigonometric Ratios handout, complete the feature map and problems 4 – 9
The Cardinal Points of the Unit Circle Review
Reminder: Special Right Triangles
23
21 2
2
30°
60°
45°
45°
11
22
30-60-90 45-45-90
Investigation!
Fit the paper triangles onto the picture below. The side with the * must be on the x-axis. Use the paper triangles to determine the coordinates of the three points.
Special Right Triangles & the Unit Circle
Special Right Triangles & the Unit Circle
Bottom half of circle
Special Right Triangles & the Unit Circle
Bottom Half of Circle
Evaluating Trigonometric Expressions
Step 1: Substitute the correct exact value for the trigonometric function. (Use the unit circle!)
Step 2: Evaluate using the order of operations
Examples2
16
sin
Examples 0cos0sin2
sin