how to split up using sum and difference identities
cos (13pi/12)
PLeaseeeeeeeeeee help!!! thanx a million
To split up the cosine function using sum and difference identities, we need to remember the following identities:
Cosine of the sum of two angles:
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
Cosine of the difference of two angles:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
Using these identities, we can split up the given cosine function:
cos(13pi/12)
We can rewrite 13pi/12 as (pi/4 + pi/3). So, we have:
cos(13pi/12) = cos(pi/4 + pi/3)
Now, applying the cosine sum identity, we get:
cos(pi/4 + pi/3) = cos(pi/4)cos(pi/3) - sin(pi/4)sin(pi/3)
To evaluate cos(pi/4), we know that it is equal to sqrt(2)/2, and sin(pi/4) is also equal to sqrt(2)/2.
Similarly, cos(pi/3) is 1/2 and sin(pi/3) is sqrt(3)/2.
Substituting these values into the equation, we have:
cos(pi/4)cos(pi/3) - sin(pi/4)sin(pi/3)
= (sqrt(2)/2) * (1/2) - (sqrt(2)/2) * (sqrt(3)/2)
= sqrt(2)/4 - sqrt(6)/4
So, the value of cos(13pi/12) is sqrt(2)/4 - sqrt(6)/4.
Remember to simplify the expression by rationalizing the denominator if required.
Hope this helps!
To split up the expression cos(13π/12) using the sum and difference identities, follow these steps:
Step 1: Recall the sum and difference identities for cosine.
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
Step 2: Identify A and B in the given expression.
A = 13π/12
B = π/12
Step 3: Calculate cos(A + B) and cos(A - B) using the sum and difference identities.
cos(13π/12) = cos(π/12 + π/12) = cos(π/12)cos(π/12) - sin(π/12)sin(π/12)
cos(13π/12) = [cos(π/6)]^2 - [sin(π/6)]^2
Step 4: Simplify the expression.
Recall the values of sin(π/6) and cos(π/6).
sin(π/6) = 1/2
cos(π/6) = √3/2
cos(13π/12) = [√3/2]^2 - [1/2]^2
cos(13π/12) = (3/4) - (1/4)
cos(13π/12) = 2/4
Step 5: Simplify the fraction.
cos(13π/12) = 1/2
Therefore, cos(13π/12) can be split up using the sum and difference identities as 1/2.
Sometimes it is easier for students to see these relationships in degrees.
13π/12 radians = 195°
and 195 = 45+150
Knowing the ratio of sides of the 30-60-90 and the 45-45-90 triangles we can find that
sin 150°=1/2
cos 150° = -√3/2
sin 45° = cos 45° = 1/√2 = √2/2
then back to
cos(13π/12)
= cos(195)
= cos(150+45)
= cos150 cos45 - sin150 sin45
= (-√3/2)(√2/2) - (1/2)(√2/2
= (-√6 - √2)/4