If sin 57=m then express the following questions
1. Sin 237
2. cos 33
3. cos 3
4. cos 66
sin 237 = sin(180+57) = -sin 57 = -m
cos 33 = cos(90-57) = sin 57 = m
cos 3 = cos(60-57) = cos60cos57+sin60sin57 = m/2 + m√3/2 = m/2(1+√3)
cos 66 = cos(60+6)
= cos60cos6 - sin60sin6
Use your double angle formulas to get sin(6) and cos(6) from cos(3) and sin(3)
To solve these questions, we can use trigonometric identities and properties.
1. Sin 237: We can use the angle addition formula for sine to find the value of sin 237.
sin(237) = sin(180 + 57)
= sin(180)cos(57) + cos(180)sin(57)
= 0(cos(57)) + (-1)(sin(57))
= -sin(57)
= -m
2. Cos 33: We can use the complementary angle identity to find the value of cos 33.
cos(33) = sin(90 - 33)
= sin(57)
= m
3. Cos 3: We can use the angle reduction formula to find the value of cos 3.
cos(3) = cos(60 - 57)
= cos(60)cos(57) + sin(60)sin(57)
= (1/2)(cos(57)) + (√3/2)(sin(57))
= (1/2)(m) + (√3/2)(sin(57))
4. Cos 66: You may already have the value for sin 57 from the given information. We can use the complementary angle identity to find the value of cos 66.
cos(66) = sin(90 - 66)
= sin(24)
→ Use the value of sin(24) to get sin(114).
= sin(90 + 24)
= sin(90)cos(24) + cos(90)sin(24)
= 1(cos(24)) + 0(sin(24))
= cos(24)
→ Use the value of cos(24) to get cos(66).
= cos(90 - 66)
= cos(90)cos(66) + sin(90)sin(66)
= 0(cos(66)) + 1(sin(66))
= sin(66)
So, for the given angles:
1. Sin 237 = -m
2. Cos 33 = m
3. Cos 3 = (1/2)(m) + (√3/2)(sin(57))
4. Cos 66 = sin(66)