1. 13-2 (Part 1): 45˚- 45 ˚- 90˚ Triangles

2. 45˚- 45˚- 90˚ Triangles
Drawing the diagonal of a square separates it into two 45˚- 45˚- 90˚ triangles.
Because it’s a square, the legs of the triangle are equal (that makes it isosceles)
Because it’s a right triangle, we can use the Pythagorean Theorem to find the length of the diagonal.

3. 45˚- 45˚- 90˚ Triangles
If the sides are of length x, then the diagonal of the square (the hypotenuse of the triangle) is:
x2 + x2 = c2
2x2 = c2
= c
To get the hypotenuse, simply multiply the side length by x x

4. 45˚- 45˚- 90˚ Triangles
Example: An official baseball diamond is a square with sides 90 ft long. How long is it from home plate to second base?
Answer:

5. 45˚- 45˚- 90˚ Triangles
Sometimes, you’ll be given the hypotenuse and asked to find the side length.
In these cases, you will divide by The thing to remember here is that you will have to rationalize your answer.

6. 45˚- 45˚- 90˚ Triangles Example of finding sides given hypotenuse.
If PQR is an isosceles right triangle, and the measure of the hypotenuse is 12, find s.

7. 45˚- 45˚- 90˚ Triangles Your Turn
Find the missing measures. Write all radicals in simplest form.
x = 10
x = 8

8. 45˚- 45˚- 90˚ Triangles Your Turn
Find the missing measures. Write all radicals in simplest form.
x = 3
y = 3

9. 45˚- 45˚- 90˚ Triangles
Assignment
Worksheet #13-2

10. 13-2 (Part 2): 30˚- 60 ˚- 90˚ Triangles

11. 30˚- 60˚- 90˚ Triangles
For 45/45/90 right triangles, we separated a square into two equal parts.
For 30/60/90 triangles, we separate equilateral triangles into two equal parts.
60˚
60˚
60˚

12. 30˚- 60˚- 90˚ Triangles Proof of the ratios up on the board.
The key to 30˚- 60˚- 90˚ triangles is to find the shortest side (the side opposite the 30˚)
60˚
30˚

13. 30˚- 60˚- 90˚ Triangles Example
In ABC, b = 7. Find a and c. Write in simplest radical form.
b is the shortest side
The hypotenuse (c) is 2x the shortest leg
c = 14
The longer leg (a) is times larger than the shortest leg
a =

14. 30˚- 60˚- 90˚ Triangles Example
In ABC, c = 18. Find a and b. Write in simplest radical form.
c is the hypotenuse
The hypotenuse is 2x the shortest leg (b)
b = 9
The longer leg (a) is times larger than the shortest leg
a =

15. 30˚- 60˚- 90˚ Triangles Your Turn In ABC, b = 8. Find a and c.
In ABC, c = 10. Find a and b.
b = 5

16. 30˚- 60˚- 90˚ Triangles Example
In ABC, a = 12. Find b and c. Write in simplest radical form.
The longer leg is times larger than the shortest leg (b)
By dividing by , we can find the short leg. Then we have to rationalize
b =
The hypotenuse (c) is 2x the shortest leg
c =

17. 30˚- 60˚- 90˚ Triangles Your Turn
In ABC, a = 8. Find b and c. Write in simplest radical form.
b =
c =

18. 30˚- 60˚- 90˚ Triangles
Assignment
Worksheet 13-3

19. 13-2 (Part 3): Angles and the Unit Circle
Essential Question: What do the sine and cosine of an angle represent on the unit circle?

20. 13-2: Angles and the Unit Circle
An angle is in standard position when the vertex is at the origin and one ray is on the positive x-axis (one angle pointing straight right).
The ray on the x-axis (pointing right) is the initial side of the angle.
The other ray is the terminal side of the angle.

21. 13-2: Angles and the Unit Circle
To measure an angle in standard position, find the amount of rotation from the initial side to the terminal side.
One full rotation contains 360˚.
How many degrees are in one quarter of a rotation?
In one half a rotation?
In three quarters of a rotation?
The angle measures 20˚ more than a straight angle of 180˚.
Since = 200, the angle is 200˚

22. 13-2: Angles and the Unit Circle
The measure of an angle is positive when the rotation is counterclockwise. The measure is negative when the rotation is clockwise.

23. 13-2: Angles and the Unit Circle
Find the measure of each angle in standard position
-315˚
-135˚
240˚
115˚
-110˚
-340˚

24. 13-2: Angles and the Unit Circle
Two angles in standard position are coterminal if they have the same terminal side
To find another coterminal angle, simply add or subtract a revolution (360˚)
There are an infinite number of angles that can be coterminal. Simply keep adding (or subtracting) revolutions…

25. 13-2: Angles and the Unit Circle
Find another angle coterminal with 198˚ by adding one full rotation.
558˚
Are angles with measures of 40˚ and 680˚ coterminal? Explain
No, 40˚ = 400˚ is coterminal
And so is 400˚ + 360˚ = 760˚
But nothing else in between is coterminal

26. 13-2: Angles and the Unit Circle
Assignment
Worksheet Practice 13-2
21 – 49 (odds)
69 – 71 (all)

27. 13-2 (Part 4): Angles and the Unit Circle
Essential Question: What do the sine and cosine of an angle represent on the unit circle?

28. 13-2: Angles and the Unit Circle
Unit Circle: A circle with a radius of 1 unit
Θ: The Greek letter “theta”, used for angle measurements

29. 13-2: Angles and the Unit Circle
Cosine of Θ (cos Θ): The x-coordinate where an angle intersects the unit circle
Sine of Θ (sin Θ): The y-coordinate where an angle intersects the unit circle
P(cos Θ, sin Θ)

30. 13-2: Angles and the Unit Circle
Example: Find the cosine and sine of 60˚
Draw a unit circle
Draw the angle indicated
Construct a right triangle
Use the rule for 45/45/90 or 30/60/90 triangles to determine the sine and cosine
60˚ 1
sin x
cos x

31. 13-2: Angles and the Unit Circle
Your Turn
Find the cosine and sine of 45˚
Find the cosine and sine of 30˚

32. 13-2: Angles and the Unit Circle
What do we do about angles greater than 90˚?
What do we do about negative angles?
The same thing… except
Be aware of negative values.
If the triangle is drawn to the left, the cosine value is negative
If the triangle is drawn to the bottom, the sine value is negative
There’s a pneumonic device we will get to after this example

33. 13-2: Angles and the Unit Circle
Example: Find the cosine and sine of -120˚
Draw a unit circle
Draw the angle indicated
Construct a right triangle
Use the rule for 45/45/90 or 30/60/90 triangles to determine the sine and cosine
60˚ 1
cos x
sin x

34. 13-2: Angles and the Unit Circle
Another trick on when sin/cos should be positive or negative
ASTC (All Students Take Calculus)
Tells you (by quadrant) which values are positive
We will get to tangent eventually, but it’s worth discussing now
Quad II
sin only
Quad I
All values
Quad III
tan only
Quad IV
cos only

35. 13-2: Angles and the Unit Circle
Assignment
Page 722
Problems 21-28
All problems
Sketches are required for credit