Without using calculator,evaluate Sin15 deg,Cos 15 deg,Sin 75deg. O need the answer urgently pls.
sin 15°
= sin(45-30)
= sin45cos30 - cos45sin30
= (1/√2)(√3/2) - (1/√2)(1/2)
= (√3 - 1)/(2√2)
so cos 15° the same way
and for the last : sin(75°) = sin(45+30)
To evaluate trigonometric functions without using a calculator, we can use the exact values for certain angles.
1. Sin 15 degrees:
To find the value of sin 15 degrees, we can use the half-angle formula for sin:
sin (x/2) = ± √[(1 - cos x)/2]
First, let's find cos 30 degrees using the exact value. Cos 30 degrees is equal to √3/2.
Now, substitute x = 30 degrees into the formula:
sin (30/2) = ± √[(1 - cos 30)/2] = ± √[(1 - √3/2)/2]
Since 15 degrees is half of 30 degrees, we substitute x = 15 degrees into the same formula:
sin (15/2) = ± √[(1 - cos 15)/2]
Now, let's find cos 15 degrees using the exact value:
cos 15 degrees = √[(1 + cos 30)/2] = √[(1 + √3/2)/2]
Substituting this value back into the half-angle formula for sin:
sin (15/2) = ± √[(1 - √[(1 + √3/2)/2])/2]
Simplifying this expression will give you the exact value of sin 15 degrees.
2. Cos 15 degrees:
We have already found cos 15 degrees using the exact value in the previous steps:
cos 15 degrees = √[(1 + √3/2)/2]
3. Sin 75 degrees:
We can use the identity sin (90 - x) = cos x to find sin 75 degrees.
sin 75 degrees = sin (90 - 15) degrees = cos 15 degrees
So, sin 75 degrees has the same value as cos 15 degrees, which is √[(1 + √3/2)/2].
By following these steps, you can evaluate sin 15 degrees, cos 15 degrees, and sin 75 degrees without using a calculator.