tangent formulas and half-angle formulas

[Questions]

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Hi,
Is it worth studying the half angle formulas? and Tangent formulas such as tan(a + b)=...?

 

[uclaDDS101]

I didn't.

 

[UCfan]

Half angle formulas are generally useless. I do not know how much detail they expect but let's assume they require for you to calculate tan (75). Let's assume you do not know the formula, but that does not mean you cannot solve the problem. Draw a triangle has 15-75-90. Then draw an imaginary line from the end of the side has 75 degree to the side close to 15 degree. In that situation you have one isosceles triangle (15-15-150) and another special triangle (30-60-90). Then look at 30-60-90 triangle and label the side sees 30 angle as a. So the side sees 60 becomes a*sqrt(3) and the side sees 90 becomes 2*a. Then by using you 15-15-150 triangle the side you dissect with you imaginary line becomes 2*a. Right now you can calculate tan 75 which is (2*a+a*sqrt(3))/a= 2+sqrt(3). As you may see you do not need to know the half angle formula in this case, even if it seems necessary to know.

 

[Questions]

how do i draw the first imaginary line you are talking about?

 

[UCfan]

how do i draw the first imaginary line you are talking about?

It does not need to be imaginary. Dissect 75 to 60 and 15. Form 15-15-150 and 30-60-90 triangles from 15-75-90. If you have further question let me know!
(Note:Look at the avatar!The big triangle tan (75) = 2+sqrt(3), or tan (15)= 1/(2+sqrt(3)). The image is not very accurate, because I used paint(not geometer sketchpad 🙂)