Expert: Josh - 12/4/2009

Question
QUESTION: hi Josh,

2 questions for you, answers supposed to be in square roots.

q1. what is the height of an equilateral traingle with side 12cm?
q2. in a right angled triangle, one angle is 45 degrees and the side next to this angle (not the hyp) has length 5cm. What is length of HYP?
answers
6root3 cm
and
5root2 cm

How do i get my answers in root form? How do i calculate when i dont have any angles to begin with in Q1?
Easy steps please.

thanks for the help

ANSWER: Hi Richard,

Remember this: The sum of all angles in a triangle is always 180 degrees.
For an equilateral triangle, all angles are equal. Therefore, each angle x=180/3.

Q1. If we look at one half of the equilateral triangle, the height is represented by AB, the angle ACB=60 (since the angle sum of triangle is 180). Side length of triangle |AC|=12.

A
|
|
|
|
B-----C

Let angle ACB be x. Using standard trigonometry, by definition, sin(x) = |AB|/|AC| (i.e., ratio between the side opposite to angle x, and the hypotenuse). Hence, the height |AB|=|AC|*sin(x)=12*sin(60)=12*(sqrt(3)/2)=6*sqrt(3).

Q2. For a right-angle triangle with one angle being 45 degrees, the remaining angle is also (180-45-90=) 45 degrees.

A
|
|
|
B-------C

Again, let angle ACB be x, angle BAC be y. Since x=45, y=180-90-x=45.
Using either sin(x)=|AB|/|AC|, or cos(x)=|BC|/|AC|, you can find the hypotenuse.
Know: i)  |AB|=|BC|=5, since angle x and y are both 45 degrees.
    ii)  sin(45)=cos(45)=1/sqrt(2)
From cos(x)=|BC|/|AC| (by definition, cosine is the ratio of the adjacent side to angle x, divided by the hypotenuse), |AC|=|BC|/cos(45)=5*sqrt(2).

Useful ratios to remember:

sin(30)=1/2
sin(45)=1/sqrt(2)
sin(60)=sqrt(3)/2

cos(30)=sqrt(3)/2
cos(45)=1/sqrt(2)
cos(60)=1/2

Deriving from these: tan(x)=sin(x)/cos(x)
tan(30)=1/sqrt(3)
tan(45)=1
tan(60)=sqrt(3)

---------- FOLLOW-UP ----------

QUESTION: Hi josh,

Q1

12*sin(60)= answer     ( got this)

sin(60)=sqrt(3)/2     (got this)

12*(sqrt(3)/2)=       6*sqrt(3).  (but how do i get to this)



q2

my answer was 5/cos45 = x = 7.1 (1d.p.)

which i understand to be 5/1/sqrt(2)

but how do i get to this       5sqrt(2)

i multiplied the fraction out   5*sqrt(2)(1) =  7.0716... = 5/sqrt(2)
which seems to be right,or i got lucky

but when i try the same way to do things with q1,,its completely wrong as im multiplying sin(60)=sqrt(3)/2 by 12
confused

how do i work out these table of values to remember them? Do I just memorize them?

multiple thanks for all the help, your advice is great, and very helpful with the other questions i don't have to ask you.
Math is a tough subject!!

Answer
I guess the confusion has to do with the way fractions are written on a single line.

Re: Q1
12*sin(60) is just 12*sqrt(3)/2, this I think you already know. [N.B. sqrt(3) represents the square root of 3] It does not matter whether we write 12*[sqrt(3)/2] as sqrt(3)*12/2 or (12/2)*sqrt(3); they are all the same. With 12 in the numerator, 2 in the denominator, 12/2 gives a multiplicative factor of 6. So, the answer is 6*sqrt(3).

Re: Q2
To see the connection between 5/[1/sqrt(2)] and 5*sqrt(2), recall that when we divide a number A by a fraction (B/C), the answer is the same as multiplying A by the reciprocal of (B/C). Since A/(B/C) = A*(C/B), when A=5, B=1 and C=sqrt(2), we have 5/[1/sqrt(2)] = 5*(sqrt(2)/1). Or simply, 5*sqrt(2).

When we evaluate 5/cos(45), 5*sqrt(2)=7.0716... is what we want. This value is same as 5/[1/sqrt(2)]=5/0.7071... but NOT 5/sqrt(2).
| sqrt(2) is roughly equal to 1.414, whereas 1/sqrt(2) is about 0.7071.

There are a number of ways of looking at this.
We can draw a 45 degree triangle, with a hypotenuse |AC|=1.

A
|
|
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B_________C

Here, |AB| and |BC| are supposed to be equal in length. So, angle ACB is 45 degrees.
Since sin(45)=|AB|/|AC| by definition, |AB| = |AC|*sin(45) = |AC|*sqrt(2).
When hypotenuse |AC|=1, the height |AB|=0.7071. It makes sense too as AB is shorter than AC.

Re: "how do i work out these table of values to remember them? Do I just memorize them?"
Ans: There are two ways to approach this.

1. Initially, to develop a feel for these ratios, you can construct a triangle geometrically, using a ruler and protractor. Let angle ACB be x. As we have done above, just draw it to scale, fix the hypotenuse |AC| to 1 and vary the angle. Then, either find out the height |AB| for sin(x), or measure the length |BC| to work out cos(x).

2. A more practical suggestion is to commit 4 things to memory.
(i)     sin(30)=1/2
(ii)    sin(45)=1/sqrt(2) ~= 0.7071 (to 4 decimal pl.)
(iii)   sin(60)=sqrt(3)/2 ~= 0.8660 (to 4 decimal pl.)
(iv)    cos(x)=sin(x+90)

This is the absolute minimum that you have to remember, if the answer is to be given in surd.
|
| DEFINITION: SURDS refer to numbers like sqrt(2) and sqrt(3)
| [generally, this includes the nth root of any integer "a", i.e., (a)^(1/n)]
| which cannot be expressed as a fraction.
|
The first three parts are commonly used ratios that you will see often in tests.
The fourth result simply tells us that sine is actually the exact waveform as cosine, except it is delayed (translated horizontally) by 90 degrees. You can compare the plot of sin(x) with cos(x) if you do a search on the web.

Part (iv) is a general and powerful result. But if it seems too complicated, you may choose to remember the following for now.
cos(30)=sqrt(3)/2       same as sin(120) and sin(60)
cos(45)=1/sqrt(2)       same as sin(135) and sin(45)
cos(60)=1/2          same as sin(150) and sin(30)
Notice the symmetry if we restrict the angle to 0 and 90 degrees.

Finally, knowing sin(x) and cos(x), we can readily work out tan(x).
e.g., tan(30)
= sin(30)/cos(30)
= (1/2)/[sqrt(3)/2]
= (1/2)*2/sqrt(3) ....the two's cancel
= 1/sqrt(3).

Yes, maths is quite a tough subject, but it can also be quite rewarding when we see the light.