Warm-Up 3/ Find the measure of
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Warm-Up 3/27-28 1. Find the measure of . sin = 14 16 =sin − 1 ( 14 16 ) =61 °
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Transcript of Warm-Up 3/ Find the measure of
Warm-Up 3/27-281. Find the measure of .
sin 𝜃=1416
𝜃=sin−1( 1416 )
𝜃=61°
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
θ
(x, y)
r y
x
sin 𝜃=¿ 𝑦𝑟
cos𝜃=¿𝑥𝑟
tan𝜃=¿ 𝑦𝑥, 𝑥≠0
𝑟=√𝑥2+𝑦2
Trigonometric Functions of Any Angle
Example 1: Let (8, –6) 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 𝑖𝑛𝜃=𝑦𝑟 ¿−610
cos𝜃=𝑥𝑟 ¿
810
ta𝑛𝜃=𝑦𝑥 ¿−68
¿− 35
¿45
¿− 34
𝑟=√82+ (−6 )2=√100𝑟=10
8
– 6 10
’
• For 0° or 360°, use the coordinate (1, 0)• For 90°, use the coordinate (0, 1)• For 180°, use the coordinate (– 1, 0)• For 270°, use the coordinate (0, – 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(– 180)
b. tan
P(– 1 , 0), r = 1
¿𝑦𝑟¿
01¿0
P(0 , – 1), r = 1
¿𝑦𝑥 ¿−10 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
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. – 150
b.
x
y
x
y
= – 150 180 – 150 = 30 ’= 30
𝜃=3𝜋4’=
4𝜋4 − 3𝜋
4 =𝜋4
θ
(x, y)
r y
x
sin 𝜃=¿𝑦𝑟
cos𝜃=¿𝑥𝑟
tan𝜃=¿𝑦𝑥, 𝑥≠0
𝑟=1sin 𝜃=𝑦cos𝜃=𝑥
(cos, sin)𝑟=√𝑥2+ 𝑦2
tan𝜃=sin 𝜃cos𝜃
x
y
Quadrant IQuadrant II
Quadrant III Quadrant IV
(+x, +y)(–x , +y)
(–x , –y) (+x , –y)
sin : cos : tan :
+++
sin : cos : tan :
+– –
sin : cos : tan :
–– +
sin : cos : tan :
–+ –
(cos, sin)
ALLStudents
Take Calculus
tan𝜃=si n 𝜃cos𝜃
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
’
12√32√33
12
√32
√3
√22√22
1
Example 4: Find the exact value of each expression.
a. cos (– 240)
x
y
’= 60
= – cos 60 ¿−12
Checkpoints:1. Find the exact values of the three basic trig functions of given (– 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
