Question

Help with 4 part question

Answer 1

let's start with 6

we can re-write sec as 1/cos and tan as sin/cos

so

sec/tan = (1/cos)/(sin/cos) = ?

(remember your rules for dividing fractions)

Answer 2

csc(x)

Answer 3

good, the equation equals 1/sin which equals csc thus, proof complete

Answer 4

Like this

Answer 5

well they might want you to break it down into steps

like, the first step could be sec(theta) = 1/cos(theta) with the reason being "sec definition" or something, then repeat for each of the substitutions we made

Answer 6

notice how we also said that tan(theta) = sin(theta)/cos(theta), that could be another step

then for the third step you could simply write sec(theta)/tan(theta) = (1/cos(theta))/(sin(theta)/cos(theta)) = 1/sin(theta) with the reasoning being "substituting for tan and sec and algebraic simplification", then write the step that converts 1/sin(theta) to csc(theta) using the csc definition

Answer 7

something like this

Answer 8

oh when I said "sec definition" I wasn't talking about the literal definition of secant, just the statement that says sec(theta) = 1/cos(theta)

Answer 9

wait so like this

Answer 10

then, next step would be to cancel out cos(theta) to get 1/sin(theta), last step would be 1/sin(theta) = csc(theta) using the definition of csc as a reason

Answer 11

anyway if you have any additional questions or just want to move onto #7 i'm ready

Answer 12

okay im ready sorry i was writing it all down

Answer 13

ok, so for 7 we would use the definitions of sec and tan to re-write sec + tan as

(1/cos(x) + sin(x)/cos(x))

using the reasoning of "combining fractions w/ common denominators" or something like that, we can re-write this as

(1+sin(x))/cos(x)

then, we can simply multiply the two fractions to get

(1+sin(x)(1-sin(x)) / [ cos(x) * cos(x) ]

this should equal 1 (check your identities if you are getting stuck)

Answer 14

okay and the reasoning for this would be the definition right

Answer 15

the definition of what

Answer 16

waot nevermind we are simply just multiplying so would there even be a reasoning for that>

Answer 17

there are several steps, let's try to break it down from the top

Answer 18

ok, so for 7 we would use the definitions of sec and tan to re-write sec + tan as

(1/cos(x) + sin(x)/cos(x))

notice how we used two definitions, one for sec, one for tan, to re-write the expression

Answer 19

yes i got that

Answer 20

after that would be combining right

Answer 21

yes, combing fractions w/ like denominators

Answer 22

then multiplying, then using the identity cos^2(x) + sin^2(x) = 1 to reduce the entire expression to 1

Answer 23

would combing fractions w/ like denominators and using the identity cos^2(x) + sin^2(x) = 1 to reduce the entire expression to 1 because different calculations or the same

Answer 24

like2 different

Answer 25

two different steps

Answer 26

like thisright =)

Answer 27

the calculations need to be on the left side, the reasonings/explanations on the right side

Answer 28

then on the left side you would write what the results of applying the steps are

Answer 29

The next reason , I am not completely sure how to calculate

Answer 30

\(\color{#0cbb34}{\text{Originally Posted by}}\) @Vocaloid
ok, so for 7 we would use the definitions of sec and tan to re-write sec + tan as

(1/cos(x) + sin(x)/cos(x))

using the reasoning of "combining fractions w/ common denominators" or something like that, we can re-write this as

(1+sin(x))/cos(x)

then, we can simply multiply the two fractions to get

(1+sin(x)(1-sin(x)) / [ cos(x) * cos(x) ]

this should equal 1 (check your identities if you are getting stuck)
\(\color{#0cbb34}{\text{End of Quote}}\)

Answer 31

what did we get after we combined the fractions?

Answer 32

good, what's the next step after combining the fractions?

Answer 33

multiplying the fractions

Answer 34

and you get (1+sin(x)(1-sin(x)) / [ cos(x) * cos(x) ]

Answer 35

good

Answer 36

then, when you expand the numerator and denominator you get

(1- sin^2(x)) / cos^2(x)

and this equals 1

thus, proof complete

Answer 37

Okay 8

Answer 38

using the secant definition

sec(x) = 1/cos(x)

therefore

sec(pi/6) = 1/cos(pi/6 - x)

using the cos difference identity cos(A-B) = cos(A)cos(B) + sin(A)sin(B)

sec(pi/6) = 1/[ cos(pi/6)cos(x) + sin(pi/6)sin(x) ]

then you just need to plug in the values of cos(pi/6) and sin(pi/6) into the equation, then do a bit of algebraic manipulation/simplification to get to the expression in the original problem

Answer 39

*plug the values from the unit circle

Answer 40

sqrt3/2 , 1/2

Answer 41

good, plug those in and simplify the expression until you get what's in the original problem

Answer 42

*sec(pi/6 - x) = 1/cos(pi/6 - x)

Answer 43

otherwise yeah, keep going

Answer 44

this is what Igot

Answer 45

the 'do a bit of algebraic manipulation" is instructions for you, not to include in the solution

Answer 46

lol oh

Answer 47

is everything else okay>

Answer 48

now, [this is instructions for you, do not include this in the answer] plug in the values for cos(pi/6) and sin(pi/6), then simplify the problem until it looks like the original equation

Answer 49

*** almost, wherever it says sec(pi/6) it needs to be sec(pi/6 - x)

Answer 50

sec(pi/6 - x)= 1/[ cos(1/2)cos(x) + sin(1/2)sin(x) ]

Answer 51

anyway it's getting pretty late here, lemme wrap this up

Answer 52

so like that

Answer 53

okay sure

Answer 54

I dont want to hold you up either :/

Answer 55

step 1: sec(pi/6 - x) = 1/cos(pi/6 - x)
reason: secant definition

step 2: 1/cos(pi/6 - x) = 1/[cos(pi/6)cos(x) + sin(pi/6)sin(x)]
reason: cos difference identity cos(A-B) = cosAcosB - sinAsinB

step 3: 1/[sqrt(3)/2cos(x) + (1/2)sin(x)]
reason: substituting the values for cos(pi/6) and sin(pi/6) from the UC

step 4: 2/[sqrt(3)cos(x) + sin(x)]
reason: multiply the numerator and denominator by 2 to eliminate the fraction

Answer 56

and done

Answer 57

9 is pretty complex but here goes:

Answer 58

tan(theta/2) = sin(theta/2)/cos(theta/2) using the tan definition

sin(theta) * 2cos(theta/2)

Answer 59

*should be sin(theta/2) in the numerator

Answer 60

anyway,

sin(theta/2) * 2 * cos(theta/2) = sin(theta) using the identity

sin(2x) = 2sin(x)cos(x) [let x = theta/2]

that's the numerator, then we move to the denominator

Answer 61

2cos^2(theta/2)

using the identity: cos^2(x) = cos(2x) + 1

gives us

2cos^2(theta/2) = cos(theta) + 1

Answer 62

putting it all together gives us

tan(theta/2) = sin(theta)/ (1 + cos(theta))

Answer 63

disclaimer: not my solution, here's the condensed version here

Answer 64

anyway I'm kinda sleepy so I'm gonna hit the hay, try to break it down step by step, showing what sort of reasoning/proof you applied for each step, then what the result of applying the step is

Answer 65

I git it... than you so much