1. Trigonometry

2. Trigonometry Non-Right Triangles Right Triangles
1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot
1. Exact values
1. Law of Sines
: AAS, ASA, SSA
2. Law of Cosines
: SAS, SSS
2. a2 + b2 = c2
3. Radian Measure of angles
3. Changing units.
4. Unit circle
5. Inverse trig functions
5. Calculator work

3. “naming the sides of the triangle.”
Right Triangles
“naming the sides of the triangle.”
What we call the legs of the triangle depend on the non-right angle given.
Sin θ =
opposite
adjacent
hypotenuse
opposite
adjacent
This is important because all of the trig functions are ratios that are defined by the lengths of these sides. For example: sine of an angle is the ratio of the length of the side opposite the angle divided by the length of the hypotenuse.

4. Confused?
adjacent
hypotenuse
opposite

5. opposite
hypotenuse
adjacent

6. Trig Functions
There are 6 trig functions we must be able to use. We must memorize their EXACT values in both radical and radian form.
Remember: trig functions are the result of ratios of the lengths of sides of a right triangle.

8. Trig Functions
There are 3 main trig functions and the 3 that are reciprocals of the first three.
The reciprocals are: Cosecant, Secant and Cotangent.
The main ones are: Sine, Cosine and Tangent.
sin θ = opp
hyp
csc θ = hyp
opp
Basically, to find the trig relationship of any angle on a right triangle, all we need to do is measure the appropriate sides of that triangle.
cos θ = adj
hyp
sec θ = hyp
adj
tan θ = opp
adj
cot θ = adj
opp
This is called “evaluating the trig functions of an angle θ.”

9. Evaluate the six trig functions of the angle θ.5
opposite 3
hypotenuse θ
adjacent 4
tan θ = 3 4
tan θ = opp
adj
sin θ = 3 5
sin θ = opp
hyp
csc θ = 5 3
cot θ = 4 3
cos θ = adj
hyp
cos θ = 4 5
sec θ = 5 4

10. We can work backwards as well
We can work backwards as well. If they give us the ratio, we can find the other trig functions.
Given:
sin θ = 5 6 θ 6
adjacent
hypotenuse
sin θ = opp
hyp
a2 + b2 = c2
csc θ = 6 5
cos θ = 6
sec θ = 6
opposite 5
sec θ = 6 11
tan θ = 5 11
tan θ = 5
cot θ = 5

11. Special Triangles: 30-60-90 and 45-45-90
30˚
45˚ 2 1
60˚
45˚ 1 1

12. θ sin θ cos θ tan θ csc θ sec θ tan θ θ sin θ cos θ tan θ csc θ sec θ
30˚
45˚
60˚ θ
sin θ
cos θ
tan θ
csc θ
sec θ
tan θ
30˚
45˚
60˚

13. Find the exact values of x and y.8 y
60˚ x

14. Find the values of x and y.16
35˚ y

15. The UNIT CIRCLE This is 1 unit long. 90˚ = π radians 2
Hence the name:
The UNIT CIRCLE 2
180˚ = π radians
360˚ = 2π radians

16. Since 180 ˚ = 1π radians we can us this as our conversion factor.
Hint: What we “want” is always in the numerator. If we want our final answer in degrees then 180 ˚ is on top. If we want radians then π radians in on top!
In other words to change
degrees into radian we multiply by π
To change
radians into degrees we multiply by π
180˚
180˚

17. Since we want radians we multiply by π/18 (radians in the numerator.
Convert 230˚ to radians.
Since we want radians we multiply by π/18
(radians in the numerator.
Which reduces to 23π 18
NO MIXED FRACTIONS!!!
230˚● π = 230π
180
180˚

18. Since we want degrees we multiply by 180/π (degrees in the numerator.)
Convert π to degrees 12
Since we want degrees we multiply by 180/π
(degrees in the numerator.)
Reduces to 15˚
Notice the π’s cancel!

19. This leads us to believe that there must be a connection between sin, cos and the coordinates (x, y)
(4, 12)
(4, 12) ●
hypotenuse
radius
opposite ● ● ●
adjacent

20. Remember the unit circle has a radius of 1 unit.
AND WE KNOW THAT THE RADIUS IN A UNIT CIRCLE IS 1 so that means:
sinθ = length of side opposite
cosθ = length of side adjacent
BUT WAIT! That’s what cos and sin are defined as!
sinθ = length of side opposite
length of hypotenuse
cosθ = length of side adjacent
So to find the coordinates of this point we can use the sin and cos if we know what the measure of the angle formed by the radius and the x axis is..
Remember the unit circle has a radius of 1 unit.
( the length of the pink line, the length of the red line)
( cos θ, sin θ ) ● θ

22. What are the coordinates of
( the length of the pink line, the length of the red line)
( cos θ, sin θ ) ● θ

23. The UNIT CIRCLE (4, 12) (4, 12) ● hypotenuse radius opposite ● ● ●
adjacent