Question

Solve the following equations over [0°, 360°].

1) 2 sin (x − 60)° = 1
2) √2 cos(x+90)°+1 = 0

Show working out thanks.

Answer 1

For question (1)
2 sin (x − 60)° = 1
Divide both sides by 2, what do you get?

Answer 2

sin(x-60)°=1/2

Answer 3

Yes. Next, take arcsine for both sides, what do you get?

Answer 4

sorry, what's arcsine?

Answer 5

opposite of sine?

Answer 6

arcsine is the inverse of sine function.

Answer 7

(x-60)°= sin-1 (1/2)

Answer 8

sin^-1

Answer 9

Yes. What does \(sin^{-1}\frac{1}{2}\) give you?

Answer 10

30

Answer 11

Yes. That's the angle in quadrant I, what about in quadrant IV?

Answer 12

360-30=330 deg

Answer 13

Or it's the good habit to memorize 1/2 = sin 30

Answer 14

I do remember it actually.

Answer 15

Oh~
So.. you get
x - 60 = 30 or x-60 = 330
Can you solve the two?

Answer 16

x=90 or x=390

Answer 17

Ah!!! My mistake!!
The other angle should be in quadrant II...
Since sine is positive in quadrant I and II.. I'm sorry!!

Answer 18

I'm sorry... Just start it again!
For question 1,
2 sin (x − 60)° = 1
Divide both sides by 2, which you've done just now.
sin (x − 60)° = 1/2
Take arcsine for both sides, you get
x-60 = 30 or x-60 = ______ (____is the angle in quadrant II, since sine of a angle in quadrant I and II gives you a positive value)

Answer 19

kk

Answer 20

Can you work out the ___ ?

Answer 21

Would it be 90?

Answer 22

Hmm....
We're still on x-60 = (angle in quadrant II), right?

Answer 23

yeh but I'm a bit confused with your method.

Answer 24

on that step.

Answer 25

how I was taught to do it was to find the angle in the first quadrant first, in this case it is 90 I believe?

Answer 26

and then using the angle that you found you use 180-that angle found= to get the answer

Answer 27

Can we do it in this way?
Solve siny = 1/2, what is/ are y?

Answer 28

y= sin^-1 (1/2)

Answer 29

which is equal to?

Answer 30

30

Answer 31

and?

Answer 32

150?

Answer 33

Right!
So, you get y=30 or y=150, agree?

Answer 34

yeh

Answer 35

Now, back you your question. 2 sin (x − 60)° = 1
sin (x-60) = 1/2
So, in this case, let x-60 = y, you'll get siny = 1/2. and just now, you solve y, which is 30 or 150. So can you get x?

Answer 36

x-60=30 or x-60=150
x=90 or x= 210

Answer 37

Now, it's done~

Answer 38

yeh but I'm still a bit confused with a part of it.

Answer 39

Which part?

Answer 40

near the last part?

Answer 41

What is it exactly?

Answer 42

"So, in this case, let x-60 = y, you'll get siny = 1/2. and just now, you solve y, which is 30 or 150. So can you get x?"

Answer 43

wait, is it possible if u type out all the working out in logical order?

Answer 44

Oh... perhaps I should write it in this way.
2sin(x-60) = 1
Let y = x-60, then the equation becomes 2siny = 1
So, siny = 1/2
And as what you did,
y= 30 or y=150
So, replace y by x-60
x-60 = 30 or x-60 = 150
Solve x.

It should be clearer now. Please ask if you still have questions. And sorry for my poor presentation just now :(

Answer 45

Or you can visualize the circle:
1/2 = sin30 = sin ( 180 - 30 )

Answer 46

kk, I get the working out but yeh I may not remember how to do it your way as I was sort of taught a different way by my cousin and I absolutely hate these type of confusing type questions.

Answer 47

It's okay. Just choose the best way for yourself. :)
Can you do question 2?

Answer 48

yep, well I'll try do it your way and I think I can only do half of the steps

Answer 49

You can try to do it using your own way first :)

Answer 50

I'll check it for you

Answer 51

wait, so I've done this so far using your method, the rest I don't get.
√2 cos(x+90)°+1 = 0
cos(x+90)° = -1/√2

let y = x + 90
therefore cos y = -1/√2
y = ............ or y = ............

Answer 52

So far so good :)
What are the values of y such that cosy = -1/√2

Answer 53

not sure

Answer 54

Hint: -1/√2 = -√2/ 2

Answer 55

Hmm.. Forget my method...
Can you solve it using your OWN way??

Answer 56

well it was my cousin's way and he only taught me just then. I still don't quite fully understand his method that's why I was planning to see if anyone could simplify it down for me on here.

Answer 57

Can you do me a favour by showing how your cousin has taught you? You can use another question or simply this question if you like.

Answer 58

tan (x+45°) = 1 x∈[0°,360°]
x+45 = 45° (x+45°) ∈ [45°,405°]

180° + 45° or 405°
x+45° = 45°, 225° or 405°
x = 0°, 180° or 360°

Answer 59

Your cousin's way is exactly as same as our way!

Answer 60

lol, really?

Answer 61

I hope you don't mind me asking the following question.
1. Have you learnt CAST ?
2. Do you know something like cos(180 - x) = -cosx
sin(180-x) = sinx and so on?
3. Are you familiar with the trigo functions of special angles (that are 0 , 30 , 45, 60, 90, and so on) ?

It's practically the same

Answer 62

yes, I have and I do know the trig functions that u mentioned

Answer 63

Oh great.
Sudden quiz. What is cos135?

Answer 64

in exact value or degrees?

Answer 65

Hint: 135 < 180

Answer 66

135 is in degrees.
cos135 gives you an exact value.

Answer 67

-cos45 = -Sqrt2/2

Answer 68

Yes.

Answer 69

Can we go back to your question 2 now?

Answer 70

yes, please. It's late over here and I want to go to sleep lol.

Answer 71

but I want to figure out how to do it properly first so I can do the other ones tomorrow.

Answer 72

Don't worry, I have to do my quiz in an hour too :|
Here's your work!
√2 cos(x+90)°+1 = 0
cos(x+90)° = -1/√2

let y = x + 90
therefore cos y = -1/√2
Can you solve y now?

Answer 73

wait. let me see... and good look for your quiz.

Answer 74

well since the value is negative it has to be in the second or third quadrant?

Answer 75

Yes. Another hint given by @Chlorophyll before : -1/√2 = - (1/√2) x (√2/√2) = - √2/2

Answer 76

from there I'm a bit stuck as to how to solve for y.

Answer 77

Consider y as X!

Answer 78

cosy = -√2/2
What is the value of angle such that cos(angle) = -√2/2?

Another hint: refer to the sudden quiz

Answer 79

yes, but how did you get the -√2/2?

Answer 80

Ratinoalization?!
-1/√2 = - (1/√2) x (√2/√2) = - √2/2

Answer 81

can't u cancel the top and bottom Sqrt 2 and so u get -1/Sqrt2

Answer 82

Oh yes, of course! But you need to know that cos45 = 1/√2

Answer 83

-√2/2 looks more familiar to you or not?

Answer 84

yes, it is cos 45 right?

Answer 85

-cos 45 in this case*

Answer 86

-cos45 = cos(___ -45) = cos(____+45) ?

Answer 87

-cos45 = cos(180-45) = cos (180+45)?

Answer 88

Yes. So...
cosy = -1/√2
y = ... or y= ...?

Answer 89

wait... let me write something else down first.

Answer 90

Take your time~

Answer 91

@JayDS @Callisto giant leap improvement here =)

Answer 92

lol, no way, I haven't improved at all but Callisto has.

Answer 93

What??!!! You haven't improved at all :'( ???

Answer 94

√2 cos(x+90)°+1 = 0
cos(x+90)° = -1/√2

let y = x + 90
therefore cos y = -1/√2
cos y = - √2/2

hmm where to from here?

Answer 95

I guess I have slightly lol, I'm just really stupid.

Answer 96

@JayDS Our goal here is to help YOU improved :)

Answer 97

No one is stupid except me, you can't be stupid than me:|
√2 cos(x+90)°+1 = 0
cos(x+90)° = -1/√2

let y = x + 90
So, the equation becomes cos y = -1/√2
Your duty: solve y. How...?!