1. Computing the Values of Trig Functions of Acute Angles
USING
SPECIAL
TRIANGLES

2. The Triangle
In a triangle the sides are in a ratio of 1- 1-
This means I can build a triangle with these lengths for sides (or any multiple of these lengths)
45°
rationalized 1
45°
90° 1

3. This means I can build a triangle with these lengths for sides
The Triangle
side opp 60°
In a triangle the sides are in a ratio of
side opp 90°
side opp 30°
This means I can build a triangle with these lengths for sides
I used the triangle and did adjacent over hypotenuse of the 60° to get this but it is the cofunction of sine so this shows again that cofunctions of complementary angles are equal.
30° 2
60°
90° 1
Be sure to locate the angle you want before you find opposite or adjacent

4. Finding the
Distance
Between
Two Points
© 2002 by Shawna Haider

5. So the distance from (-6,4) to (1,4) is 7.
Let's find the distance between two points.
7 units apart 8 7 6
So the distance from (-6,4) to (1,4) is 7. 5 4
(-6,4)
(1,4) 3 2 1 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 -2 -3 -4 -5 -6 -7
If the points are located horizontally from each other, the y coordinates will be the same. You can look to see how far apart the x coordinates are.

6. So the distance from (-6,4) to (-6,-3) is 7.
What coordinate will be the same if the points are located vertically from each other? 8 7 6 5 4
(-6,4) 3 2
7 units apart 1 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 -2 -3
So the distance from (-6,4) to (-6,-3) is 7.
(-6,-3) -4 -5 -6 -7
If the points are located vertically from each other, the x coordinates will be the same. You can look to see how far apart the y coordinates are.

7. The Pythagorean Theorem will help us find the hypotenuse
But what are we going to do if the points are not located either horizontally or vertically to find the distance between them? 8 7
Let's start by finding the distance from (0,0) to (4,3) 6 5 4 5 3 ? 2 3 1 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 -2 4 -3
The Pythagorean Theorem will help us find the hypotenuse -4 -5 -6
So the distance between (0,0) and (4,3) is 5 units. -7
This triangle measures 4 units by 3 units on the sides. If we find the hypotenuse, we'll have the distance from (0,0) to (4,3)
Let's add some lines and make a right triangle.

8. Again the Pythagorean Theorem will help us find the hypotenuse
Now let's generalize this method to come up with a formula so we don't have to make a graph and triangle every time. 8
Let's start by finding the distance from (x1,y1) to (x2,y2) 7
(x2,y2) 6 5 ? 4
y2 – y1 3
(x1,y1) 2 1 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8
x2 - x1 -2 -3
Again the Pythagorean Theorem will help us find the hypotenuse -4 -5 -6 -7
Solving for c gives us:
Let's add some lines and make a right triangle.
This is called the distance formula

9. 3 -5 Let's use it to find the distance between (3, -5) and (-1,4)
Plug these values in the distance formula
(x1,y1)
(x2,y2)
means approximately equal to -1 3 4 -5
CAUTION!
found with a calculator
Don't forget the order of operations!
You must do the parenthesis first then powers (square the numbers) and then add together BEFORE you can square root