warm-up 3/ find the measure of
TRANSCRIPT
Warm-Up 3/27-281. Find the measure of .
sin ๐=1416
๐=sinโ1( 1416 )
๐=61ยฐ
Rigor:You will learn how to find reference
angles, to evaluate and determine sign of trig functions of any angle.
Relevance:You will be able to solve real world problems using reference angles.
Trig 3: Trigonometric
Functions on the Unit Circle
ฮธ
(x, y)
r y
x
sin ๐=ยฟ ๐ฆ๐
cos๐=ยฟ๐ฅ๐
tan๐=ยฟ ๐ฆ๐ฅ, ๐ฅโ 0
๐=โ๐ฅ2+๐ฆ2
Trigonometric Functions of Any Angle
Example 1: Let (8, โ6) be a point on the terminal side of and angle in standard position. Find the exact values of the three basic trig functions of .
s ๐๐๐=๐ฆ๐ ยฟโ610
cos๐=๐ฅ๐ ยฟ
810
ta๐๐=๐ฆ๐ฅ ยฟโ68
ยฟโ 35
ยฟ45
ยฟโ 34
๐=โ82+ (โ6 )2=โ100๐=10
8
โ 6 10
โ
โข For 0ยฐ or 360ยฐ, use the coordinate (1, 0)โข For 90ยฐ, use the coordinate (0, 1)โข For 180ยฐ, use the coordinate (โ 1, 0)โข For 270ยฐ, use the coordinate (0, โ 1 )
2
32
0 or 2r = 1
Example 2: Find the exact value of each trigonometric function, if defined. If not defined, write undefined.
a. sin(โ 180)
b. tan
P(โ 1 , 0), r = 1
ยฟ๐ฆ๐ยฟ
01ยฟ0
P(0 , โ 1), r = 1
ยฟ๐ฆ๐ฅ ยฟโ10 ๐ข๐๐๐๐๐๐๐๐
Reference Angle: an acute angle formed by the terminal side and the x-axis.
Example 3: Sketch each angle. Then find the reference angle.
a. โ 150
b.
x
y
x
y
= โ 150 180 โ 150 = 30 โ= 30
๐=3๐4โ=
4๐4 โ 3๐
4 =๐4
ฮธ
(x, y)
r y
x
sin ๐=ยฟ๐ฆ๐
cos๐=ยฟ๐ฅ๐
tan๐=ยฟ๐ฆ๐ฅ, ๐ฅโ 0
๐=1sin ๐=๐ฆcos๐=๐ฅ
(cos, sin)๐=โ๐ฅ2+ ๐ฆ2
tan๐=sin ๐cos๐
x
y
Quadrant IQuadrant II
Quadrant III Quadrant IV
(+x, +y)(โx , +y)
(โx , โy) (+x , โy)
sin : cos : tan :
+++
sin : cos : tan :
+โ โ
sin : cos : tan :
โโ +
sin : cos : tan :
โ+ โ
(cos, sin)
ALLStudents
Take Calculus
tan๐=si n ๐cos๐
2
32
2
EVALUATING TRIG FUNCTIONS AT ANY ANGLE
1.Find the reference angle.2.Find the corresponding trig value.3.Determine the sign of the angle.
x
y
O
โ
12โ32โ33
12
โ32
โ3
โ22โ22
1
Example 4: Find the exact value of each expression.
a. cos (โ 240)
x
y
โ= 60
= โ cos 60 ยฟโ12
Checkpoints:1. Find the exact values of the three basic trig functions of given (โ 4 , 3).
r = 5,
2. Find the exact value of the trig function .
r = 1,
3. Sketch the angle and then find the reference angle of 300.
4. Find the exact value of the trig function .
Quadrant III tan is positive.
Assignment4-3 Worksheet 1-19 all
Warm-Up 3/28
1. Find the exact value of the trig function .
r = 1,
2. Find the exact value of the trig function .
Quadrant II cos is negative.
Assignment4-3 Worksheet 1-19 all