Evaluate sin 7pi/12. Using sum, difference, and cofunction identities
To evaluate sin(7π/12) using sum, difference, and cofunction identities, we can utilize the following identities:
1. sine of a sum formula: sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
2. sine of a difference formula: sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
3. cofunction identity: sin(π/2 - A) = cos(A)
Let's break down the process step by step:
Step 1: Identify angles that you can use with the above identities. In this case, we have 7π/12 as our angle.
Step 2: Rewrite the given angle (7π/12) as a sum or difference of angles that we can use with the identities. We can rewrite 7π/12 as (π/6 + π/4). Therefore, we have sin(7π/12) = sin(π/6 + π/4).
Step 3: Use the sine of a sum formula to rewrite sin(π/6 + π/4) in terms of trigonometric functions of π/6 and π/4:
sin(π/6 + π/4) = sin(π/6)cos(π/4) + cos(π/6)sin(π/4).
Step 4: Apply the cofunction identity to rewrite sin(π/6) as cos(π/2 - π/6):
sin(π/6)cos(π/4) + cos(π/6)sin(π/4) = cos(π/2 - π/6)cos(π/4) + cos(π/6)sin(π/4).
Step 5: Simplify the expression using known values:
cos(π/6) = √3/2
cos(π/4) = √2/2
sin(π/4) = √2/2
Substituting the values, we have:
cos(π/2 - π/6)cos(π/4) + cos(π/6)sin(π/4) = (√3/2)(√2/2) + (√3/2)(√2/2).
Step 6: Evaluate the expression:
(√3/2)(√2/2) + (√3/2)(√2/2) = (√6/4) + (√6/4) = 2√6/4 = √6/2.
Therefore, sin(7π/12) = √6/2.
well, 7 = 6 + 1
so it is really
sin (6 pi/12 + pi/12)
sin(a+b) = sin a cos b + cos a sin b
(just like your other problem)
sin 6 pi/12 = sin 90 = 1
cos pi/12 = cos 15 deg
cos 90 = 0
so cos 15 degrees