Additional Math – Trigonometry – How to derive Cosine Half Angle.

Use the Cosine Double Angle Formula (found in Formula List) to help derive Cosine 1/2 Angle.

$latex \displaystyle \text{cos}\ 2A=2{{\cos }^{2}}A-1$

Substitute A for 1/2 x
$latex \displaystyle \text{cos}\ 2\left( {\frac{1}{2}x} \right)={{\cos }^{2}}\left( {\frac{1}{2}x} \right)-1$

Simplify the equation (*Note: you can multiply cos 2(1/2x) to get cos x.)
$latex \displaystyle \text{cos}\ x={{\cos }^{2}}\left( {\frac{1}{2}x} \right)-1$
$latex \displaystyle \text{1+cos}\ x={{\cos }^{2}}\left( {\frac{1}{2}x} \right)$
$latex \displaystyle \sqrt{{\text{1+cos}\ x}}=\cos \left( {\frac{1}{2}x} \right)$
$latex \displaystyle \cos \left( {\frac{1}{2}x} \right)=\sqrt{{\text{1+cos}\ x}}$


There is a debate as to whether Half-Angle formulas  are part of the N or O Level Exam syllabus. I ask my students and two of my Math teacher friends,the answer was inconclusive, some students said that they are taught in school, others say that they never seen this in their entire life, even the teachers were not sure. I check the SEAB (Singapore Examination and Assessment Board) website but there was no info (explicit or implicit) about this. The funny thing is that half-angle formulas do appear in textbooks , some schools worksheet and test papers. In the next post i will show you how to derive sine 1/2 angle formula.

The Bottom Line: Better to err on the side of caution

Additional Math Tuition for students living in Choa Chu Kang, Yew Tee,Sembawang and Yishun.


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