how to do the calculation while taking 'A'
as 15 in the half angle formula used for calculating the value of sin 7.5 ?
Unless you have found sin15 or cos15 before, and it has become part of your trig repertoire , you will have to do it twice
since you must know that cos 30 = √3/2
and using
cos 2A = 2cos^2 A - 1 , let 2A = 30
cos30 = 2cos^2 15 - 1
solving this for cos 15 I got √(√3 + 2)/2
now use the other expansion for cos 2A
cos 2A = 1 - 2sin^2 A
cos 15 = 1 - 2sin^2 7.5
this gave me
2sin^2 7.5 = 1 - √(√3 + 2)/2
sin 7.5 = √[2 - √(√3 + 2) ]/2
you can check it with sin 7.5 on your calculator, it is correct.
To calculate the value of sin 7.5 using the half-angle formula, you first need to find the value of sin A/2, where A is 15.
Here are the steps to follow:
1. Convert the angle A from degrees to radians. Since the half-angle formula uses radians, we need to convert the angle from degrees to radians. The conversion formula is: Radians = Degrees * (π / 180). So, for A = 15 degrees, the value of A in radians will be: A_radians = 15 * (π / 180).
2. Divide the angle A_radians by 2 to find A/2. In this case, A/2 = 15 * (π / 180) / 2.
3. Use a calculator or a trigonometric table to find the value of sin(A/2). Plug in the value of A/2 calculated above into the sin function: sin(A/2).
For example, if you have a scientific calculator, follow these steps:
a. Press the "Mode" key and make sure the calculator is set to radian mode.
b. Enter the value of A/2 into the calculator: sin(A/2).
c. Press the "=" key to get the result.
The result will give you the value of sin(7.5) based on the given angle A = 15.