I like to write the radian forms in degrees so it is easier to find a sum/difference that I can use for a relevant angle addition template. That way, you won't have to mess with fractions.
-π/12 = -15 degrees. (Treat π as 180 when converting)
You can choose 30 and 45 as your a and b, since 30 - 45 = -15 so write it as cos(30 - 45)
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I like to write the radian forms in degrees so it is easier to find a sum/difference that I can use for a relevant angle addition template. That way, you won't have to mess with fractions.
-π/12 = -15 degrees. (Treat π as 180 when converting)
You can choose 30 and 45 as your a and b, since 30 - 45 = -15 so write it as cos(30 - 45)
cos(a - b) = cosAcosB + sinAsinB
= cos30cos45 + sin30sin45
= (√3/2)(√2/2) + (1/2)(√2/2)
= √6/4 + √2/4
= (√6 + √2) / 4
Use half-angle identity: cos²x = 1/2 (1 + cos(2x))
cos²(−π/12) = 1/2 (1 + cos(−π/6))
cos²(−π/12) = 1/2 (1 + √3/2)
cos²(−π/12) = 1/4 (2+√3)
cos²(−π/12) = (4+2√3)/8
cos²(−π/12) = (1+√3)²/8
Since −π/12 is in Q4, cos(−π/12) > 0, so take positive square root:
cos(−π/12) = (1+√3)/(2√2) = (√2+√6)/4
Well, you know that cos(π/12) = cos(-π/12)
and that π/12 = (1/2) (π/6)
so. use the half-angle identity (you know tht cos(π/6)=(√3)/2)
or use the difference identity (cos(-π/12) = cos(π/4 - π/3))
Either works...
Use the cos (a - b) = cos a cos b + sin a sin b identity
where a = 3π/12 (π/4) and b = 4π/12 (π/3)