Question

5(cos^2)60 + 4(sec^2)30 - (tan^2)45 = x
Find x

Answer 1

well cos(60) and sec(30) and tan(45) can all be evaluated using the unit circle

Answer 2

http://buildingaunitcircle.weebly.com/uploads/6/2/2/3/622393/9409299.gif

Answer 3

Use the trig table........

Answer 4

do you need help using that chart above ?

Answer 5

....
I need to the values
like cos = 1/2
and all.......

Answer 6

cos(60)=1/2 that is correct
so squaring both sides gives
cos^2(60)=1/4

Answer 7

sec(30) means you need to find cos(30) and flip whatever number that is

Answer 8

tan(45) means you need to find sin(45) and cos(45)
but both sin(45) and cos(45) are the same so tan(45)=1

Answer 9

\[5 \cos^2(60)+4\sec^2(30)-\tan^2(45) \\ 5(\frac{1}{4})+4(\frac{1}{\cos(30)})^2-1\]

Answer 10

so you still need to find cos(30)

Answer 11

4/3 = sec^2(30)

Answer 12

\[\cos(30)=\frac{\sqrt{3}}{2} \\ \cos^2(30)=\frac{3}{4} \\ \frac{1}{\cos^2(30)}=\frac{4}{3} \\ \sec^2(30)=\frac{4}{3}\]
that is right

so far you have this now:
\[5(\frac{1}{4})+4(\frac{4}{3})-1\]

Answer 13

that can be simplify just be doing a few multiplications and additions(subtractions)

Answer 14

\[\frac{ 5 }{ 4 } +\frac{ 16 }{ 3 } - 1\]

Answer 15

right so far

Answer 16

shall I take LCM?

Answer 17

yes which the lcm(4,3,1)=12

Answer 18

\[\frac{5}{4}+\frac{16}{3}-\frac{1}{1} \\ \frac{5(3)}{4(3)}+\frac{16(4)}{3(4)}-\frac{1(12)}{1(12)} \\ \frac{5(3)}{12}+\frac{16(4)}{12}-\frac{12}{12} \\ \frac{5(3)+16(4)-12}{12}\]

Answer 19

67/12

Answer 20

looks awesome!

Answer 21

and totally correct :)

Answer 22

Thanks!
Please help me a bit more?

Answer 23

Sure. I can take a stab at it.

Answer 24

@freckles would using identities just take longer?

Answer 25

@triciaal if you want to try something with identities you can

Answer 26

\[\frac{ cotA-1 }{ 2 -\sec^2(A) } = \frac{ \cot A }{ 1 + tanA }\]

Answer 27

some people don't need to put in terms of sin and cos
but I always do because I remember more identities with sin and cos

Answer 28

Help me?

Answer 29

\[\frac{\frac{\cos(a)}{\sin(a)}-1}{2-\frac{1}{\cos^2(a)}} =\frac{\frac{\cos(a)}{\sin(a)}}{1+\frac{\sin(a)}{\cos(a)}}\]
so this is what I would do as a first step
second step would be to remove the compound fraction action going on

Answer 30

I'm suppose to use LHS and get RHS

Answer 31

ok then get rid of the compound fraction action on the the left hand side

Answer 32

\[\frac{ \cos(a) - \sin(a) }{ \sin(a) }\]

Answer 33

and for the denominator
2sin^(a)/cos^2(a)

Answer 34

\[\frac{\frac{\cos(a)}{\sin(a)}-1}{2-\frac{1}{\cos^2(a)}} \cdot \frac{\sin(a) \cos^2(a)}{\sin(a)\cos^2(a)} \\ \frac{\cos^3(a)-\cos^2(a)\sin(a)}{2 \cos^2(a)\sin(a)-\sin(a)} \\ \text{ so doing a bit of factoring } \\ \frac{\cos^2(a)(\cos(a)-\sin(a))}{\sin(a)(2 \cos^2(a)-1)} \\ \\ \text{ now recall on the other side we wanted } \cot(a) \\ \text{ on \top } \\ \text{ well we notice } \frac{\cos(a)}{\sin(a)}=\cot(a) \\ \cot(a) \frac{\cos(a)(\cos(a)-\sin(a))}{2 \cos^2(a)-1} \\ \]

now somehow we have to show that
\[\frac{\cos(a)(\cos(a)-\sin(a))}{2 \cos^2(a)-1}=\frac{1}{1+\tan(a)}\]
any ideas @BlackDranzer

Answer 35

if not read this if you want spoilers:
\[\frac{\cos(a)(\cos(a)-\sin(a))}{2 \cos^2(a)-1} \cdot \frac{\cos(a)+\sin(a)}{\cos(a)+\sin(a)} \\ \frac{\cos(a)(\cos^2(a)-\sin^2(a))}{(2 \cos^2(a)-1)(\cos(a)+\sin(a))} \\ \text{ well recall double angle identity for } cosine \\ \frac{ \cos(a)}{\cos(a)+\sin(a)}\]
now divide both top and bottom by cos(a)

Answer 36

that is I used \[\cos(2a)=\cos^2(a)-\sin^2(a) \\ \text{ and also } \cos(2a)=2\cos^2(a)-1\]

Answer 37

yep and we can bring down that cot(a) factor we had from earlier
(though I normally write my fractions like this 1/(1+tan(a))
)

Answer 38

so the key in our proof was the use of double angle identity for cosine

Answer 39

Nice..
Thanks...
I need help in quadratic equations too...

Answer 40

ok let's see it

Answer 41

hey @triciaal did you come up with a way using identities for the first one ?

Answer 42

I honestly haven't given it a try

Answer 43

An aero plane flying at 1 km horizontally above the ground make an angle of elevation 60 after 10 seconds its 30 find the speed of the aerplane in km/hr

Answer 44

wait what does that mean
what is the angle of elevation 60 or 30?

Answer 45

and what is 30?

Answer 46

from 60 to 30 in secs
from the smae point

Answer 47

oh the angle of elevation changed from 60 to 30 in 10 sec
gotcha