Evaluate
*Note - We have to find the exact value of these. That I know to do. For example sin5π/12 will be broken into sin (π/6) + (π/4)
So...
sin 5π/12
sin (π/6) + (π/4)
sin π/6 cos π/4 + cos π/6 sin π/4
I get all those steps. The part I am struggling with is breaking these into 2 simple ones.
a) sin 11π/12
b) cos 13π/12
c) tan (-7π/12)
d) tan (-5π/12)
11 is 5 + 6
-7 is -12 + 5
-5 is -2 -3
13 is 2 + 11, and 11 is above.
Watch the algebra.
5 and 6 over what?
They have to either be over 3, 4, 6, or 2.
To evaluate these trigonometric expressions, we can follow a similar process as the one you mentioned. We'll break them into simpler trigonometric expressions using known identities and values.
a) Evaluating sin 11π/12:
To break this down into simpler expressions, we'll use the identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b).
sin 11π/12 = sin (π/6 + π/4)
We can now apply the identity:
sin (π/6 + π/4) = sin π/6 cos π/4 + cos π/6 sin π/4
Recall that sin π/6 and cos π/6 are known values (1/2 and √3/2 respectively), and sin π/4 and cos π/4 are also known values (√2/2).
sin 11π/12 = (1/2)(√2/2) + (√3/2)(√2/2)
Multiply and simplify:
sin 11π/12 = √2/4 + √6/4
This is the exact value of sin 11π/12.
b) Evaluating cos 13π/12:
Similarly, we'll use the identity cos (a + b) = cos(a)cos(b) - sin(a)sin(b).
cos 13π/12 = cos (π/6 + π/4)
Now applying the identity:
cos (π/6 + π/4) = cos π/6 cos π/4 - sin π/6 sin π/4
Using the known values, cos π/6 = √3/2, cos π/4 = √2/2, sin π/6 = 1/2, and sin π/4 = √2/2:
cos 13π/12 = (√3/2)(√2/2) - (1/2)(√2/2)
Multiply and simplify:
cos 13π/12 = √6/4 - √2/4
This is the exact value of cos 13π/12.
c) Evaluating tan (-7π/12):
To evaluate tan (-7π/12), we can use the identity tan(-θ) = -tan(θ).
tan (-7π/12) = -tan (7π/12)
Then, we'll use the identity tan (a + b) = (tan(a) + tan(b))/(1 - tan(a)tan(b)).
tan (7π/12) = (tan (π/6 + π/4))/(1 - tan (π/6)tan (π/4))
Using the known values, tan (π/6) = 1/√3 and tan (π/4) = 1:
tan (7π/12) = (1/√3 + 1)/(1 - (1/√3)(1))
Multiply and simplify:
tan (7π/12) = (√3 + 1)/(√3 - 1)
This is the exact value of tan (-7π/12).
d) Evaluating tan (-5π/12):
Following the same process as above, we have:
tan (-5π/12) = -tan (5π/12)
tan (5π/12) = (tan (π/6 + π/4))/(1 - tan (π/6)tan (π/4))
Using the known values, tan (π/6) = 1/√3 and tan (π/4) = 1:
tan (5π/12) = (1/√3 + 1)/(1 - (1/√3)(1))
Multiply and simplify:
tan (5π/12) = (√3 + 1)/(√3 - 1)
This is the exact value of tan (-5π/12).
Remember to simplify or rationalize the denominators if required in order to get the final answer.