Download Right-angled triangles, Pythagoras and Trigonometry a + b = h ( or c

Right-angled triangles, Pythagoras and Trigonometry
Here is the line ____‘a’
Here is a
2
This is a right-angle or ninety-degree angle (90 )
All of these triangles are ‘right angled’ triangles.
Pythagoras discovered many hundreds of years ago, that if you want to find the length of a missing side on a triangle you can do so by looking at the SQUARES of the sides first, and then work
backwards.
He found that the side opposite the
right angle, which he called the
hypotenuse (h) , if squared, always
contained the SUM of the other 2 sides
if each of them were also squared.
2
2
2
2
a + b = h ( or c )
32 + 42 = h2
3x3 + 4x 4 = h2
9 + 16 = 25
h2 = /25
h=5
If you were given ‘b’ and ‘c’ but not ‘a’, you
could find the missing side by flipping about
your equation.
b2 = h2 - a 2
or
a 2 = h 2 - b2
2
2
2
b =c -a
or
a 2 = c 2 - b2
Then he worked backwards to find the
actual missing side by reducing the
square back to a single line or nonsquared measurement.
Count the small squares in the sum or
hypotenuse squared.
Do the other small squares from side a
and side b, fit inside it?
What if you wanted to find the measurement of the other angles in the triangle? Or, you were
only given one side, and an angle and had to find the missing side?
Years ago, the measurements or ratios of every possible angle related to every possible
side were calculated by mathematicians and put into detailed trigonometry charts or ‘tables‘.
These same ratios are now stored in your calculators. The three ratios we use most are sine, cosine
and tangent. (or sin, cos and tan for short). The ratios consist of sides or angles that are opposite
what you are trying to find, or near it (adjacent) , or opposite the right angle —the hypotenuse,
remember?
sin = opposite cos = adjacent
tan = opposite
hypotenuse
hypotenuse
adjacent
How do you know which formula to use?
1. You label all of the missing sides.
2. Then you look to see what numbers you have.
3. Next, you ‘plug in’ the numbers.
1. Label the sides, related to the angle you’ve
been given and the side you want to find.
3. You have an ‘a’ but no ‘o’ or ‘h’and only one
other angle. So the best formula to use would be
toh (tan)
t=o
tan 55 = o now flip the formula
a
140
O = tan 55 x 140 = 200 ( press tan button, 55x140)
1. Label your angle.
2. Label the sides, related to the angle you
want to find.
3. You have an ‘a’ and an ‘h’ but not an ‘o’ so the best
formula to use would be Cah (cos 1)
c=a
h
How would you find other missing sides or angles at this point?
Once you have 2 sides, you can always find the third using your
Pythagorean Theorem.
How would you find the other angles?
Once you have 2 angles you can find the third by subtracting
the sum of those 2 angles from 180.
or (angle) < d = 6 = .75 shift button, cos button
8
= 41.41