Why sin(30) = 1/2

In trigonometry, sometimes we require students to learn the special values of sine, cosine and tangent at 30, 45, 60 degrees, because these are nice values. But I wonder if these facts are explained or do teachers just say "sin(30) = 1/2 , memorise this!" 

Here's my explanation.(I am not sure if they teach this in schools, but I'll look up the textbook the next time I visit popular bookstore.)

Show that sin(30) = Opp/ Hyp = 1/2.

30-Triangle.jpg

1) Draw a new line dividing the right-angle into 30 and 60 degrees respectively.

30-Triangle2.jpg

Now triangle ABD and BCD are both isosceles. In fact, BCD is an equilateral triangle.

2) Hence,  CD= BD = BC

3) Since BD lies on the isosceles triangle ABD, BD= AD.

4) Conclude sin(30) = opp/hyp = BC / AC = 1/2

5) For good measure, using pythagoras theorem, we can now show that

T1.gif

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16 Comments

  1. Sean Vianale said,

    October 7, 2008 at 6:39 am

    Thank You So Much!!!!!!
    I was for 2 hours figuring out why Sin(30)=1/2 and now you have showed me. This really helped me much. Good Job also.

  2. RON said,

    May 21, 2009 at 12:53 am

    I learned this in the ninth grade, and was the ratio of one line called the opposite side and the hypotenuse. But it was understood about what you show on paper against what teachers at the time did not elaborate on. Some times old folks know that all teachers are created equal.

  3. November 14, 2009 at 11:36 pm

    I learned this in the Class X.

  4. deepak pandey said,

    April 1, 2010 at 12:49 pm

    Draw an equilateral triangle ABC.

    Draw a perpendicular from A to BC intersecting BC at D
    AB = AC (sides of an equilateral trianlge are equal)
    Angle ADB = Angle ADC = 90°
    AD = AD
    ΔADB ≅ ΔADC (RHS postulate)

    BD = DC = (1/2)BC = (1/2)AB (congruence of triangles and sides of an equilateral triangle are equal)

    Angle ABD = AngleDAC = 60°/2 = 30° (congruence of triangles)

    sinBAD = BD/AB = 1/2
    sin(30°) = 1/2

  5. Gurbir Singh said,

    June 1, 2010 at 10:54 pm

    Thanks a lot.
    With regards
    Gurbir Singh

  6. sameer said,

    August 25, 2010 at 5:21 pm

    sir your rule is valid only for special case but i can give idea to find the valu of any angle

  7. sameer said,

    August 27, 2010 at 9:16 pm

    how the trigonometry will be genrated who is responsible
    “ans”=pythagoras
    because he invented right angle triangle
    hence we know l^2+b^2=h^2
    now divide both side by h^2
    this employed sin^2a +cos^2a=1
    now we can genrate all trgonometric formulla

  8. sameer said,

    August 27, 2010 at 9:25 pm

    why sin30=1/2 ?
    let x=30,
    3x=90,
    take sin of both side,
    sin3x=1,
    3sinx-4sin^3x=1,
    4sin^3x-3sinx+1=0,
    sinx=1/2,1/2,-1;

    but sinx =-1employes x=270
    hence sinx=1/2,
    put valu of x=30
    hence make all angles as multiple of 90

  9. sameer said,

    August 27, 2010 at 9:28 pm

    if you having any problem related to maths then consult me i will try to help you

  10. satyendra said,

    September 15, 2010 at 1:00 pm

    0 is even or odd numbetr

  11. sameer mishra said,

    November 27, 2010 at 3:01 pm

    o is even or odd number

  12. Lanston said,

    April 26, 2011 at 9:14 pm

    Great! my school teacher always say:”Just bury them in mind”,But I found my answers here, Thanks a lot,Hope there’re more articles here

  13. Lavlesh mishra said,

    May 2, 2011 at 2:43 am

    thank you sir ,i was very confused and at that time yore help is very important for me

  14. D P said,

    June 19, 2013 at 3:01 pm

    Thanks U Sir…

  15. gary said,

    January 29, 2014 at 12:09 pm

    thanks very much..awesome explanation

  16. shekhar sharma said,

    July 17, 2014 at 12:07 am

    Its not too good explanation. I am not agree to this solution. Its too typical to make the students understand cosec 30 or sec 30 etc


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