# cosine rule

• Nov 25th 2006, 03:31 AM
jim49990
cosine rule
could any one help please..yet to be taught the cosine rule but trying to learn it myself but un sure what buttons to press on my calculator..
i have a triangle A, B,F. <A=52 side f=3000 and b=6000..i need to find the other side and two angles..
thanks
jim
• Nov 25th 2006, 04:22 AM
CaptainBlack
Quote:

Originally Posted by jim49990
could any one help please..yet to be taught the cosine rule but trying to learn it myself but un sure what buttons to press on my calculator..
i have a triangle A, B,F. <A=52 side f=3000 and b=6000..i need to find the other side and two angles..
thanks
jim

The cosine rule for your triangle is:

a^2=f^2+b^2-2fc cos(A),

since everything on the right hand side is a known you can find
a fairly easily.

Now the sine rule will allow you to find the remaining angles:

a/sin(A)=b/sin(B)=f/sin(F).

RonL
• Nov 25th 2006, 04:34 AM
jim49990
so would that give me a answer of 10190.00
thanks
• Nov 25th 2006, 04:44 AM
CaptainBlack
Quote:

Originally Posted by jim49990
so would that give me a answer of 10190.00
thanks

a^2=3000^2+6000^2-2*3000*6000 cos(52),

a^2=(9+36-36x0.61566)*1000^2=30.2232 1000^2

so:

a=sqrt(22.836)*1000~=4778.72

RonL
• Nov 25th 2006, 05:13 AM
jim49990
still stuck
sorry to be a pain but i am still struggling on what buttons to press on my calculator
jim
:confused:
• Nov 25th 2006, 06:02 AM
CaptainBlack
Quote:

Originally Posted by jim49990
sorry to be a pain but i am still struggling on what buttons to press on my calculator
jim
:confused:

Don't try to do it in one go on your calculator, they all differ and may not
behave in the same way as someone elses calculator.

Find 3000^3, and 6000^2, 3000*6000 and cos(52), these are single stage
operations then write out what what you now have to calculate.

So cos(52)=0.615661 (clear the calculator, make sure you are in degrees
mode, then enter 52 and hit the cos button to get this, write it down).

3000^2=9e6 (which means 9 * 1000000, clear the calculator enter 3000
then hit the x^2 key, or enter x 3000 =)
6000^2=3.6e7 (which means 36 *1000000, as above)

3000*6000=1.8e7 (which means 18 * 1000000)

then calculate:

9e6+3.6e7-2*3.6e7*0.61661=22836.204

then hit the square root key to get 4778.7..

RonL
• Nov 25th 2006, 09:27 AM
jim49990
many thanks
cheers mate..i owe you a drink
jim
• Nov 26th 2006, 01:53 AM
jim49990
angles
would <f =29 degrees
thanks
• Nov 26th 2006, 06:43 AM
CaptainBlack
Quote:

Originally Posted by jim49990
would <f =29 degrees
thanks

I make it 29.62 degrees.

Unsing the sine rule you should have:

sin(F)/f=sin(A)/a,

so

sin(F)=f*sin(A)/a=3000*sin(52)/4778.2=0.627851*0.78801=0.494752

and asin(0.494752)~=29.62 degrees

RonL
• Nov 26th 2006, 10:10 AM
Soroban
Hello, Jim!

Quote:

Yet to be taught the cosine rule but trying to learn it myself
but unsure what buttons to press on my calculator.

I have a triangle $\displaystyle ABF$ with: $\displaystyle A = 52^o,\;f = 3000,\;b = 6000$
I need to find the other side and two angles.

To find side $\displaystyle a\!:\; a^2\;=\;b^2 + f^2 - 2bf\cos A$

We have: .$\displaystyle a^2\;=\;6000^2 + 3000^2 - 2\!\cdot\!6000\!\cdot\!3000\cos52^o$

Be sure your calculator is in degree mode.

. . $\displaystyle 6000\;\;\boxed{x^2}\;\;\boxed{+}\;\;3000\;\;\boxed {x^2}\;\;\boxed{-}\;\;2\;\;\boxed{\times}\;\;6000\;\;\boxed{\times} \;\;3000\;\;\boxed{\times}\;\;\boxed{\cos}\;\;52\; \;\boxed{=}$

It is essential that you press $\displaystyle \boxed{=}$ at the end of this line.

. . Then press: .$\displaystyle \boxed{\land}\;\;0.5\;\;\boxed{=}$

You should get: .$\displaystyle a\;=\;4778.722307\quad\Rightarrow\quad\boxed{a\;\a pprox\;4778.7}$

To find angle $\displaystyle B:\;\frac{\sin B}{b} \,=\,\frac{\sin A}{a}\quad\Rightarrow\quad \sin B \,=\,\frac{b\sin A}{a}$

We have: .$\displaystyle \sin B\:=\:\frac{6000\sin52^o}{4778.7}$

Enter: .$\displaystyle 6000\;\;\boxed{\times}\;\;\boxed{\sin}\;\;52\;\;\b oxed{)}\;\;\boxed{\div}\;\;4778.7\;\;\boxed{=}$ . . . . and get: $\displaystyle 0.989403922$

Press $\displaystyle \boxed{\sin^{-1}}\;\;\boxed{\text{Ans}}\;\;\boxed{=}$ . . . . and get: $\displaystyle 81.65176939$

Hence: .$\displaystyle B \:\approx\:81.65^o\quad\hdots\quad \text{or: }B \:\approx\:98.35^o$ **

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If $\displaystyle B = 81.65^o$, then: .$\displaystyle F \:=\:180^o - 52^o - 81.65^o \:=\:46.35^o$

We have: .$\displaystyle \boxed{\begin{array}{ccc}A\:= \\ B\:= \\ F\;=\end{array} \begin{array}{ccc}52^o \\ 81.65^o \\ 46.35^o\end{array} \begin{array}{ccc}a\:= \\ b\:= \\ f\:=\end{array} \begin{array}{ccc}4778.7 \\ 6000 \\ 3000\end{array}}$

But this doesn't check out . . .

. . $\displaystyle \frac{a}{\sin A} \:=\:\frac{4778.7}{\sin52^o} \:\approx\:6064.257$

. . $\displaystyle \frac{b}{\sin B} \:=\:\frac{6000}{\sin81.35^o} \:\approx\:6064.284$

. . $\displaystyle \frac{f}{\sin F}$ $\displaystyle \:=\:\frac{3000}{\sin46.35^o}\:\approx\:4146.110$ ??

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**

Whenever we use $\displaystyle \boxed{\sin^{-1}}$, there are always two possible answers.
One is the acute angle on your calculator screen,
. . the other is its supplement (subtract from 180°)

The calculator said $\displaystyle B = 81.65^o$, which didn't work.
. . So it must be: .$\displaystyle B \:=\:180^o - 81.65^o \:=\:98.35^o$

If $\displaystyle B = 98.35^o$, then $\displaystyle F\:=\:180^o - 52^o - 98.35^o \:=\:29.65^o$

The solution is: .$\displaystyle \boxed{\begin{array}{ccc}A\:=\\B\:=\\F\:=\end{arra y} \begin{array}{ccc}52^o\\98.35^o\\29.65^o\end{array } \begin{array}{ccc}a\:=\\b\:=\\f\:=\end{array} \begin{array}{ccc}4778.7 \\ 6000 \\ 3000\end{array}}$

. . and these check out . . .