## JayDS 3 years ago Solve the following equations over [0°, 360°]. 1) 2 sin (x − 60)° = 1 2) √2 cos(x+90)°+1 = 0 Show working out thanks.

1. Callisto

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

2. JayDS

sin(x-60)°=1/2

3. Callisto

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

4. JayDS

sorry, what's arcsine?

5. JayDS

opposite of sine?

6. Callisto

arcsine is the inverse of sine function.

7. JayDS

(x-60)°= sin-1 (1/2)

8. JayDS

sin^-1

9. Callisto

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

10. JayDS

30

11. Callisto

12. JayDS

360-30=330 deg

13. Chlorophyll

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

14. JayDS

I do remember it actually.

15. Callisto

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

16. JayDS

x=90 or x=390

17. Callisto

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

18. Callisto

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)

19. JayDS

kk

20. Callisto

Can you work out the ___ ?

21. JayDS

Would it be 90?

22. Callisto

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

23. JayDS

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

24. JayDS

on that step.

25. JayDS

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?

26. JayDS

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

27. Callisto

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

28. JayDS

y= sin^-1 (1/2)

29. Callisto

which is equal to?

30. JayDS

30

31. Callisto

and?

32. JayDS

150?

33. Callisto

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

34. JayDS

yeh

35. Callisto

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?

36. JayDS

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

37. Callisto

Now, it's done~

38. JayDS

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

39. Callisto

Which part?

40. JayDS

near the last part?

41. Callisto

What is it exactly?

42. JayDS

"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?"

43. JayDS

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

44. Callisto

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 :(

45. Chlorophyll

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

46. JayDS

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.

47. Callisto

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

48. JayDS

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

49. Callisto

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

50. Callisto

I'll check it for you

51. JayDS

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 = ............

52. Callisto

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

53. JayDS

not sure

54. Chlorophyll

Hint: -1/√2 = -√2/ 2

55. Callisto

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

56. JayDS

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.

57. Callisto

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.

58. JayDS

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°

59. Chlorophyll

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

60. JayDS

lol, really?

61. Callisto

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

62. JayDS

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

63. Callisto

Oh great. Sudden quiz. What is cos135?

64. JayDS

in exact value or degrees?

65. Chlorophyll

Hint: 135 < 180

66. Callisto

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

67. JayDS

-cos45 = -Sqrt2/2

68. Callisto

Yes.

69. Callisto

Can we go back to your question 2 now?

70. JayDS

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

71. JayDS

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

72. Callisto

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?

73. JayDS

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

74. JayDS

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

75. Callisto

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

76. JayDS

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

77. Chlorophyll

Consider y as X!

78. Callisto

cosy = -√2/2 What is the value of angle such that cos(angle) = -√2/2? Another hint: refer to the sudden quiz

79. JayDS

yes, but how did you get the -√2/2?

80. Callisto

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

81. JayDS

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

82. Callisto

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

83. Chlorophyll

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

84. JayDS

yes, it is cos 45 right?

85. JayDS

-cos 45 in this case*

86. Callisto

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

87. JayDS

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

88. Callisto

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

89. JayDS

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

90. Callisto

91. Chlorophyll

@JayDS @Callisto giant leap improvement here =)

92. JayDS

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

93. Callisto

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

94. JayDS

√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?

95. JayDS

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

96. Chlorophyll

97. Callisto

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...?!

98. Callisto

*delete ''you can't be stupid than me''

99. JayDS

haha thanks, I guess I'm just a logical person, like I things shown out clearly well in a way that I can understand of course and I'm slow at learning, well depends on what type of maths as well.

100. JayDS

I like*

101. Chlorophyll

@Callisto wanna me to take over so you have time with your quiz?

102. JayDS

sorry for taking up your time guys and thanks for your help so far.

103. Callisto

@Chlorophyll Ah... I forget my quiz :| I'll come back as soon as possible!!

104. Chlorophyll

@Callisto Good luck, Sis! Not to be worried I promise to a good job here :)

105. Callisto

@Chlorophyll Sis, I trust you :) Thanks too!!

106. JayDS

what's with the "Sis" lol

107. Chlorophyll

√2 cos(x+90)°+1 = 0 -> cos(x+90)° = -1 /√2 = -√2/2 Since cos value is negative, now replace ... into where -√2/2 is, can you!

108. JayDS

sorry, I'm confused about the replacing part.

109. Chlorophyll

Hint: the angle in the second and third quadrants

110. JayDS

yep, I know that but... I'm unsure how to use that

111. JayDS

second quadrant = pi - angle so it will be pi - sqrt2 /2 third quadrant = pi + angle so it will be pi + Sqrt 2/2 Is that correct?

112. Chlorophyll

cos(x+90)° = -√2/2 cos(x+90)° = -cos45 = cos(180-45) = cos (180+45)

113. Chlorophyll

So you have 2 cases here!

114. Chlorophyll

x+90° = 180-45 and .....

115. JayDS

yeh, that is the same as what I said earlier? pi - sqrt 2/2 and pi + sqrt 2/2 I believe.

116. JayDS

but then you would need to convert is back to degrees.

117. Chlorophyll

It's incorrect to connect the degree ( pi) with the radian ( arc length ) pi - sqrt 2/2

118. JayDS

kk, can you quickly explain this line "cos(x+90)° = -cos45 = cos(180-45) = cos (180+45)"

119. JayDS

oh wait, I think I got it.

120. Chlorophyll

Don't pay attention to the left part yet!

121. Chlorophyll

-√2/2 = -cos45 Now we wish to get rid of negative sign: -cos45 = cos ( 180 - 45) -cos45 = cos ( 180 + 45)

122. Chlorophyll

That's what Callisto has been working so hard to show you the concept!

123. JayDS

√2 cos(x+90)°+1 = 0 cos(x+90)° = -1/√2 let y = x + 90 So, the equation becomes cos y = -1/√2 cos y = -√2/2 cos y = -cos 45 cos (x+90) = -cos 45 cos45 = cos ( 180 - 45) cos45 = cos ( 180 + 45)

124. Chlorophyll

Actually now you've done with the right side, plug it into the equation: cos (x+90) = -cos 45 In the second quadrant: cos (x+90) = cos ( 180 - 45) -> x+90 = ....?

125. JayDS

x + 90 = 135???

126. Chlorophyll

-> x = ...?

127. JayDS

45

128. Chlorophyll

Similarly for the third quadrant case, can you do it yourself?

129. JayDS

well if x=45 wouldn't it just be 180+45=225?

130. Chlorophyll

cos (x+90) = cos ( 180 + 45) -> x+90 = ....?

131. JayDS

x + 90 = 225 x = 135

132. Chlorophyll

I suggest that you don't jump around, as we carefully show you step by step, just follow it one by one, and practice it yourself from now on!

133. JayDS

LOL, okay big sis ^^ I will practice tomorrow but it's getting late so I'm going to sleep now.

134. Chlorophyll

Okie, the solution already saved into your profile. Feel free to use it as your study note!

135. JayDS

really, how?

136. JayDS

is that a new function?

137. Chlorophyll

138. Chlorophyll

139. JayDS

Maybe you can only do because you have already reached 99 smart points or something.

140. JayDS

SmartScore*

141. Chlorophyll

No, you can check anybody profiles!

142. JayDS

yeh kk, found it, it was on the left side, I didn't see it over there and I thought you couldn't click anything there.

143. JayDS

Well I g2g now Sis, thanks so much for the help and help me say thanks to Callisto as well.