find the exact value of sin 112.5 in fraction form
112.5 = 90 + 22.5
ah ha, 22.5 is half of 45
we will be able to do this with half angle formulas
first:
sin 122.5 = sin(90+22.5) = sin 90 cos 22.5 + cos 90 sin 22.5 = cos 22.5 + 0
so we want cos (45/2)
cos 45/2 = sqrt[ (1+cos 45)/2]
= sqrt [ (1 +1/sqrt2)/2 }
= sqrt [ (1 + sqrt 2) /2 sqrt 2 ]
= sqrt [ (2 + sqrt 2) /4 ]
= (1/2)sqrt (2+sqrt 2)
about .923879 which checks
112.5 = 5π/8 = (5π/2)/4
so, use your half-angle formula twice:
cos(x/2) = √((1+cos(x))/2)
cos(5π/4) = -√((1+0)/2) = -1/√2
since 5π/4 is in QIII
sin(x/4) = √((1-cos(x/2))/2)
sin(5π/8) = √((1-cos(5π/4)/2)
= √((1+1/√2)/2) = √((√2+1)/2√2)
good insight, simpler solution.
Shows that with trig, there is almost always more than one path to solutions.
To find the exact value of sin 112.5° in fraction form, we can use the sum-to-product identities.
First, we note that 112.5° is the sum of two angles, 45° and 67.5°.
The sum-to-product identity for sine states that sin(A + B) = sin(A) * cos(B) + cos(A) * sin(B).
Using this identity, we can rewrite sin 112.5° as sin(45° + 67.5°).
Now, we know that sin 45° = 1/√2 and cos 45° = 1/√2, because it is a special angle.
Similarly, sin 67.5° is not a special angle, so we can't find its exact value easily.
However, we know that sin 67.5° = cos (90° - 67.5°) = cos 22.5°.
Using the same logic as before, we find that cos 22.5° = 1/√2 + 1/√2.
Now we can substitute these values into the sum-to-product identity:
sin 112.5° = sin(45° + 67.5°)
= sin 45° * cos 67.5° + cos 45° * sin 67.5°
= (1/√2) * (1/√2 + 1/√2).
Simplifying, we have:
sin 112.5° = (1/√2) * (2/√2)
= 2/2
= 1.
Therefore, the exact value of sin 112.5° in fraction form is 1.