Question

Two step equations

Answer 1

first use the sin sum and difference formulas to re-write sin(x+pi/4) and sin(x-pi/4)

Answer 2

sin(A+B) = sin(A)cos(B) + cos(A)sin(B)

sin(x+pi/4) = sin(x)cos(pi/4) + cos(x)sin(pi/4)

try to apply the same kind of reasoning for sin(x-pi/4)

Answer 3

sin(x-pi/4) = sin(x)cos(pi/4) - cos(x)sin(pi/4)

Answer 4

awesome

now putting this together we get

sin(x)cos(pi/4) + cos(x)sin(pi/4) + sin(x)cos(pi/4) - cos(x)sin(pi/4) = 1

see anything you can simplify/cancel out?

Answer 5

+ cos(x)sin(pi/4)

Answer 6

sin(x)cos(pi/4) + sin(x)cos(pi/4) = 1

Answer 7

whoops made a mistake, should have been a minus sign

sin(x)cos(pi/4) + cos(x)sin(pi/4) - [ sin(x)cos(pi/4) - cos(x)sin(pi/4) ] = 1

therefore

2cos(x)sin(pi/4) = 1

^ try solving this for x

Answer 8

sqrt 2

Answer 9

first find what sin(pi/4) equals using the unit circle

then solve for cos(x)

then figure out what the value of x must be

Answer 10

sqrt 2 cos x =1

Answer 11

good, so cos(x) = 1/sqrt(2) = sqrt(2)/2

what values of x does this correspond to?

Answer 12

1/sqrt2

Answer 13

if cos(x) = sqrt(2)/2 what must x equal?

Answer 14

pi/4 and 7pi/4

Answer 15

awesome so x = pi/4 and 7pi/4 are your solutions

Answer 16

for B)

sin(2x) = 2cos(x)sin(x)

try to apply this to

sin(x)cos(x) = sqrt(3)/2

to re-write the expression in terms of sin only

Answer 17

do i use the chart you provided

Answer 18

hint:

try solving sin(2x) = 2cos(x)sin(x) for cos(x)sin(x)

then re-write the original equation in terms of sin only

Answer 19

\[\sin(2x)=2\sin^2(x)\cos^2(x)\]

Answer 20

i think :/

Answer 21

if 2cos(x)sin(x) = sin(2x)

then what does cos(x)sin(x) equal?

Answer 22

hint: divide both sides by 2.

Answer 23

cos(x)sin(x) = sin(x)

Answer 24

ugh had a brain fart

Answer 25

you cannot divide the 2x inside of the sin expression

2cos(x)sin(x) = sin(2x)

dividing both sides by 2 gives us

cos(x)sin(x) = (1/2)sin(2x)

now go back to the original expression and substitute this inside to re-write the original expression in terms of sin only

Answer 26

sin(x)cos(x) = sqrt(3)/4

sin(x)cos(x) = (1/2)sin(2x)

therefore:

(1/2)sin(2x) = sqrt(3)/4

solve for x

Answer 27

multiply both sides of the equation by 2 to get

sin(2x) = sqrt(3)/2

look on the unit circle where sinx = sqrt(3)/2 and then divide those solution(s) by 2 because it's sin(2x) not sinx

Answer 28

pi /6 and pi/3

Answer 29

good

Answer 30

for C) try factoring

tan^2(x) - 3tan(x) + 2 = 0

if you are having trouble, try letting a = tan(x) and re-write the equation in terms of a

Answer 31

(tanx -2)(tanx-1) is what it should factor to, set that equal to 0 and solve for the appropriate tan values

Answer 32

x=1.11

Answer 33

x=pi/4

Answer 34

good but also x = 5pi/4 (that's where sin and cos both equal -sqrt(2)/2)