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

1. Callisto Group Title

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

2. JayDS Group Title

sin(x-60)°=1/2

3. Callisto Group Title

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

4. JayDS Group Title

sorry, what's arcsine?

5. JayDS Group Title

opposite of sine?

6. Callisto Group Title

arcsine is the inverse of sine function.

7. JayDS Group Title

(x-60)°= sin-1 (1/2)

8. JayDS Group Title

sin^-1

9. Callisto Group Title

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

10. JayDS Group Title

30

11. Callisto Group Title

12. JayDS Group Title

360-30=330 deg

13. Chlorophyll Group Title

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

14. JayDS Group Title

I do remember it actually.

15. Callisto Group Title

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

16. JayDS Group Title

x=90 or x=390

17. Callisto Group Title

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 Group Title

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 Group Title

kk

20. Callisto Group Title

Can you work out the ___ ?

21. JayDS Group Title

Would it be 90?

22. Callisto Group Title

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

23. JayDS Group Title

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

24. JayDS Group Title

on that step.

25. JayDS Group Title

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 Group Title

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

27. Callisto Group Title

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

28. JayDS Group Title

y= sin^-1 (1/2)

29. Callisto Group Title

which is equal to?

30. JayDS Group Title

30

31. Callisto Group Title

and?

32. JayDS Group Title

150?

33. Callisto Group Title

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

34. JayDS Group Title

yeh

35. Callisto Group Title

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 Group Title

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

37. Callisto Group Title

Now, it's done~

38. JayDS Group Title

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

39. Callisto Group Title

Which part?

40. JayDS Group Title

near the last part?

41. Callisto Group Title

What is it exactly?

42. JayDS Group Title

"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 Group Title

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

44. Callisto Group Title

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 Group Title

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

46. JayDS Group Title

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 Group Title

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

48. JayDS Group Title

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

49. Callisto Group Title

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

50. Callisto Group Title

I'll check it for you

51. JayDS Group Title

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 Group Title

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

53. JayDS Group Title

not sure

54. Chlorophyll Group Title

Hint: -1/√2 = -√2/ 2

55. Callisto Group Title

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

56. JayDS Group Title

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 Group Title

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 Group Title

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 Group Title

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

60. JayDS Group Title

lol, really?

61. Callisto Group Title

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 Group Title

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

63. Callisto Group Title

Oh great. Sudden quiz. What is cos135?

64. JayDS Group Title

in exact value or degrees?

65. Chlorophyll Group Title

Hint: 135 < 180

66. Callisto Group Title

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

67. JayDS Group Title

-cos45 = -Sqrt2/2

68. Callisto Group Title

Yes.

69. Callisto Group Title

Can we go back to your question 2 now?

70. JayDS Group Title

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

71. JayDS Group Title

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

72. Callisto Group Title

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 Group Title

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

74. JayDS Group Title

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

75. Callisto Group Title

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

76. JayDS Group Title

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

77. Chlorophyll Group Title

Consider y as X!

78. Callisto Group Title

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

79. JayDS Group Title

yes, but how did you get the -√2/2?

80. Callisto Group Title

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

81. JayDS Group Title

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

82. Callisto Group Title

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

83. Chlorophyll Group Title

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

84. JayDS Group Title

yes, it is cos 45 right?

85. JayDS Group Title

-cos 45 in this case*

86. Callisto Group Title

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

87. JayDS Group Title

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

88. Callisto Group Title

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

89. JayDS Group Title

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

90. Callisto Group Title

91. Chlorophyll Group Title

@JayDS @Callisto giant leap improvement here =)

92. JayDS Group Title

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

93. Callisto Group Title

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

94. JayDS Group Title

√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 Group Title

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

96. Chlorophyll Group Title

97. Callisto Group Title

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 Group Title

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

99. JayDS Group Title

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 Group Title

I like*

101. Chlorophyll Group Title

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

102. JayDS Group Title

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

103. Callisto Group Title

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

104. Chlorophyll Group Title

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

105. Callisto Group Title

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

106. JayDS Group Title

what's with the "Sis" lol

107. Chlorophyll Group Title

√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 Group Title

sorry, I'm confused about the replacing part.

109. Chlorophyll Group Title

Hint: the angle in the second and third quadrants

110. JayDS Group Title

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

111. JayDS Group Title

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 Group Title

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

113. Chlorophyll Group Title

So you have 2 cases here!

114. Chlorophyll Group Title

x+90° = 180-45 and .....

115. JayDS Group Title

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

116. JayDS Group Title

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

117. Chlorophyll Group Title

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

118. JayDS Group Title

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

119. JayDS Group Title

oh wait, I think I got it.

120. Chlorophyll Group Title

Don't pay attention to the left part yet!

121. Chlorophyll Group Title

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

122. Chlorophyll Group Title

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

123. JayDS Group Title

√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 Group Title

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 Group Title

x + 90 = 135???

126. Chlorophyll Group Title

-> x = ...?

127. JayDS Group Title

45

128. Chlorophyll Group Title

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

129. JayDS Group Title

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

130. Chlorophyll Group Title

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

131. JayDS Group Title

x + 90 = 225 x = 135

132. Chlorophyll Group Title

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 Group Title

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

134. Chlorophyll Group Title

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

135. JayDS Group Title

really, how?

136. JayDS Group Title

is that a new function?

137. Chlorophyll Group Title

138. Chlorophyll Group Title

139. JayDS Group Title

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

140. JayDS Group Title

SmartScore*

141. Chlorophyll Group Title

No, you can check anybody profiles!

142. JayDS Group Title

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 Group Title

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