1. Radian and Degree Measure

2. Trigonometry:
Greek word meaning “measurement of triangles.” It deals with relationships among the sides and angles of triangles. So the first applications were used in astronomy, navigation, and surveying.
In the 17th century, the development of calculus and the physical sciences led to viewing trigonometric relationships as functions with the set of real numbers as their domains. This then expanded the use of trigonometry to include physical phenomena involving rotations and vibrations. These phenomena include sound waves, light rays, planetary orbits, vibrating strings, pendulums, and orbits of atomic particles.
Everything we will study this semester leads to the capability to further your education in the studies of math and science.

3. Angles:
An angle is determined by rotating a ray about its endpoint. The starting ray is the initial side, and ending ray is the terminal side. The endpoint of the ray is the vertex. y x
An angle in standard position I T
Terminal side
Initial side
Vertex
On a coordinate system, an angle is in standard position if the vertex is at the origin and the initial side is on the positive x-axis.

4. Positive, Negative, and Coterminal angles:
Positive angles are generated by counterclockwise rotation.
Negative angles are generated by clockwise rotation.
Coterminal angles are angles that have the same initial and terminal sides. (to find coterminal angles, we add or subtract 2𝜋) y x
Positive angle
(counterclockwise) y x
Coterminal
Angles - A & B A B
Negative angle
(clockwise)

5. Radian Measure: One way to measure angles is in radians. The formula
**For all mathematics from this point forward, almost all angle measures are done in Radians.**
The formula
𝜃= 𝑠 𝑟 θ=𝑟𝑎dians, s=arc length, and r=radius
describes the relationship
between parts of a circle
and the radian angle measure. 𝜃 𝑟 y x 𝑠

6. Radians: We know that 𝐶=2𝜋𝑟
Then when the radius of a circle is 1, the number of radians in a circle is 2𝜋.
Since 2𝜋≈6.28, then there are 6.28 radians in a circle. y x
1 𝑟𝑎𝑑𝑖𝑎𝑛
2 𝑟𝑎𝑑𝑖𝑎𝑛s
3 𝑟𝑎𝑑𝑖𝑎𝑛s
6 𝑟𝑎𝑑𝑖𝑎𝑛s
5 𝑟𝑎𝑑𝑖𝑎𝑛s
4 𝑟𝑎𝑑𝑖𝑎𝑛s

7. Quadrants and Key Angles:
Key Angles we will use:
1 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛=2𝜋
1 2 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛=𝜋
1 4 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛= 𝜋 2
1 6 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛= 2𝜋 6 = 𝜋 3
1 8 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛= 2𝜋 8 = 𝜋 4
1 12 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛= 2𝜋 12 = 𝜋 6
𝜃= 𝜋 2 (90°)
𝑄𝑢𝑎𝑑𝑟𝑎𝑛𝑡 𝐼𝐼
𝜋 2 <𝜃<𝜋
𝑄𝑢𝑎𝑑𝑟𝑎𝑛𝑡 𝐼
0<𝜃< 𝜋 2
𝜃=𝜋 (180°)
𝜃=0 (0°)
𝑄𝑢𝑎𝑑𝑟𝑎𝑛𝑡 𝐼𝐼𝐼
𝜋<𝜃< 3𝜋 2
𝑄𝑢𝑎𝑑𝑟𝑎𝑛𝑡 𝐼𝑉
3𝜋 2 <𝜃<2𝜋
𝜃= 3𝜋 2 (270°)

8. Complements and Supplements:
Two positive angles are complementary if
their sum is 𝜋 ° .
Two positive angles are supplementary if
their sum is π 180° .
Note: an angle may not have a complement or supplement if it is too big.
Example: 5𝜋 3 is too big to have either a complement or supplement ( 𝑖𝑠 𝑏𝑖𝑔𝑔𝑒𝑟 𝑡ℎ𝑎𝑛 𝑎𝑛𝑑 1)

9. Examples:
Sketch an angle in standard position:
1) 7𝜋 6
2) − 𝜋 3
3) 7𝜋 2
4) −4π

10. Examples:
Determine the quadrant that each angle is in:
5) 5𝜋 6
6) − 𝜋 3
7) −300°

11. Examples:
Find the complement and supplement (if possible):
9) 72°
10) 𝜋 3
11) 5𝜋 6
12) 2

12. Examples:
Find two coterminal angles (1 positive and 1 negative):
13) 120°
14) 2𝜋 3
15) − 𝜋 4
16) − 31𝜋 6

13. Degrees- Radians Conversion:
We know that 2𝜋=360°, so 𝜋=180°.
Degrees to radians, multiply by 𝜋 180
Radians to degrees, multiply by 180 𝜋
Answers:
1) 3𝜋 4
2) −3𝜋 2
3) 150°
4) °
Examples:
1) 135°
2) −270°
3) 5𝜋 6
4) 2 rad.

14. 𝑻𝒉𝒆 𝒇𝒐𝒍𝒍𝒐𝒘𝒊𝒏𝒈 𝒇𝒐𝒓𝒎𝒖𝒍𝒂𝒔 𝒘𝒐𝒓𝒌 𝒂𝒔 𝒍𝒐𝒏𝒈 as 𝜽 𝒊𝒔 𝒎𝒆𝒂𝒔𝒖𝒓𝒆𝒅 𝒊𝒏 𝒓𝒂𝒅𝒊𝒂𝒏𝒔
Formulas:
𝑻𝒉𝒆 𝒇𝒐𝒍𝒍𝒐𝒘𝒊𝒏𝒈 𝒇𝒐𝒓𝒎𝒖𝒍𝒂𝒔 𝒘𝒐𝒓𝒌 𝒂𝒔 𝒍𝒐𝒏𝒈 as
𝜽 𝒊𝒔 𝒎𝒆𝒂𝒔𝒖𝒓𝒆𝒅 𝒊𝒏 𝒓𝒂𝒅𝒊𝒂𝒏𝒔 𝜃 𝑟 𝑠
Arc Length:
𝑠=𝑟𝜃 r=radius,θ=central angle
Linear Speed:
v= 𝑠 𝑡 s=arc length, t=𝑡𝑖𝑚𝑒
Angular Speed:
w= 𝜃 𝑡 θ=central angle, t=𝑡𝑖𝑚𝑒
Area of a Sector of a Circle: A= 1 2 𝑟 2 𝜃 θ=central angle, r=radius θ=central angle, r=radius

15. Angles you will need to know going forward:
𝜋 2 (90°)
0 (0°)
𝜋 (180°)
3𝜋 2 (270°)
𝜋 6 (30°)
𝜋 4 (45°)
𝜋 3 (60°)

16. Homework: pp , #’s 25-76