Download Section 7-3 The Sine and Cosine Functions

Warm Up
A right triangle has side lengths x,
y, and r. Find the unknown length.
2.
𝑥 = 4, 𝑦 = 5 𝒓 = 𝟒𝟏
𝑥 = 21, 𝑟 = 29 𝒚 = 𝟐𝟎
3.
𝑟 = 10, 𝑥 = 1
𝒚=𝟑
4.
𝑦 = 4 2, 𝑥 = 4
𝒓=𝟒 𝟑
1.
5.
The square has side lengths 14. The two
1
curves are each of a circle with radius 14.
4
Find the area of the shaded region.
r
y
x
𝟗𝟖𝝅 − 𝟏𝟗𝟔
Quiz 7.1 & 7.2
Degree
& Radian Conversions
Coterminal Angles
Arc Length of a Sector
Area of a Sector
Apparent Size
Section 7-3
The Sine and Cosine Functions
Objective: To use the definitions of
sine and cosine to find values of
these functions and to solve simple
trigonometric equations.
𝑠𝑖𝑛𝜃 =
𝑜𝑝𝑝
𝑦
=
ℎ𝑦𝑝
𝑟
cos 𝜃 =
𝑎𝑑𝑗
𝑥
=
ℎ𝑦𝑝
𝑟
r

Example 1
If the terminal ray of an angle θ in standard
position passes through (-3, 2), find sin θ and
cos θ.
Solution:
On a grid, locate (-3,2).
Use this point to draw a right triangle, where one
side is on the x-axis, and the hypotenuse is line
segment between (-3,2) and (0,0).
If the terminal ray passes through (  3,2), find
sin & cos .
𝑦
2
2 13
𝑠𝑖𝑛𝜃 =
=
=
𝑟
13
13
−3
𝑥
−3 13
𝑐𝑜𝑠𝜃 = =
=
𝑟
13
13
𝒙 = −𝟑
𝐲=𝟐
𝑟 = −3
𝒓 = 𝟏𝟑
2
+ 2
2
Example 2
5
5
= − , what
quadrant
is the,
If If the
is a𝑠𝑖𝑛𝜃
4th Quadrant

and
sin


13
13
angle in?
find cos .
5
If  is a 4th Quadrant  and sin   ,
13
find cos .
𝑥 12
𝑐𝑜𝑠𝜃 = =
𝑟 13
𝐲 = −𝟓
𝒓 = 𝟏𝟑
𝑥=
2
13
𝒙 = ±𝟏𝟐
− −5
2
4th Quadrant, so
𝒙 = 𝟏𝟐
𝑠𝑖𝑛𝜃 =
𝑜𝑝𝑝
𝑦
=
ℎ𝑦𝑝
𝑟
cos 𝜃 =
𝑎𝑑𝑗
𝑥
=
ℎ𝑦𝑝
𝑟
r

When the radius =1 on the unit circle,
𝑦
𝑠𝑖𝑛𝜃 = = 𝑦
1
𝑥
𝑐𝑜𝑠𝜃 = = 𝑥
1
Unit Circle
The circle x2 + y2 = 1 has radius 1 and is therefore
called the unit circle. This circle is the easiest one
with which to work because sin θ and cos θ are
simply the y- and x-coordinates of the point where
the terminal ray of θ intersects the circle.
When the radius =1 on the unit circle,
𝑦
𝑠𝑖𝑛𝜃 = = 𝑦
1
𝑥
𝑐𝑜𝑠𝜃 = = 𝑥
1
1
2
1
1
2
On Your Unit Circle:
 Label the quadrants.
 Note the positive or negative x and y values in each
quadrant.
(cos, sin)
(cos, sin)
(−, +)
(+, +)
II
I
III IV
(−, −)
(cos, sin)
(+, −)
(cos, sin)
You can determine the exact value of sine
and cosine for many angles on the unit
circle. Find:
A. sin
90°
B. sin 450°
C. cos (-π)
D. sin
2𝜋
(− )
3
E. cos
-315°
A. 1
B. 1
C. -1
D. −
E.
2
2
3
2
Example 3
Solve sin θ = 1 for θ in degrees and radians.
Degrees: 𝜃 = 90˚ ± 360𝑛
𝜋
2
Radians: 𝜃 = ± 2𝑛𝜋
Repeating Sin and Cos Values
For any integer n,
𝑠𝑖𝑛 (𝜃 ± 360°𝑛) = 𝑠𝑖𝑛 𝜃
𝑐𝑜𝑠 (𝜃 ± 360°𝑛) = 𝑐𝑜𝑠 𝜃
𝑠𝑖𝑛 (𝜃 ± 2𝜋𝑛) = 𝑠𝑖𝑛 𝜃
𝑐𝑜𝑠 (𝜃 ± 2𝜋𝑛) = 𝑐𝑜𝑠 𝜃
The sine and cosine functions are periodic.
They have a fundamental period of 360˚ or 2
radians.
Homework
Page 272
#1-27 odd, #33-41 odd