Use compound angle formulas to evaluate
a) cos(pi/12)
b) sin(pi/12)
c) tan(5pi/12)
d) sin(11pi/12)
just apply the half-angle formulas.
Post your work if you get stuck.
for the first one i got [sqrt(2)(1+sqrt(3))]/4 . can i simplify this further?
not really. I might say
(1+√3)/√8 or (1+√3)/(2√2)
To evaluate the trigonometric functions using compound angle formulas, we need to express the given angles in terms of known angles (usually π/6, π/4, or π/3) and then use the corresponding formulas. Let's evaluate each of the given trigonometric functions:
a) Evaluating cos(π/12):
To express π/12 in terms of known angles, we can write it as π/6 - π/4.
Using the formula for the cosine of a difference of angles: cos(A - B) = cos(A)cos(B) + sin(A)sin(B), we can substitute A = π/6 and B = π/4 to get:
cos(π/12) = cos(π/6)cos(π/4) + sin(π/6)sin(π/4)
The values of cos(π/6), cos(π/4), sin(π/6), and sin(π/4) are known. Substitute these values and simplify the expression:
cos(π/12) = (√3/2)(√2/2) + (1/2)(√2/2)
= (√6 + √2)/4
Therefore, cos(π/12) = (√6 + √2)/4.
b) Evaluating sin(π/12):
To express π/12 in terms of known angles, we can write it as π/6 - π/4.
Using the formula for the sine of a difference of angles: sin(A - B) = sin(A)cos(B) - cos(A)sin(B), we can substitute A = π/6 and B = π/4 to get:
sin(π/12) = sin(π/6)cos(π/4) - cos(π/6)sin(π/4)
The values of sin(π/6), cos(π/4), cos(π/6), and sin(π/4) are known. Substitute these values and simplify the expression:
sin(π/12) = (1/2)(√2/2) - (√3/2)(√2/2)
= (√2 - √6)/4
Therefore, sin(π/12) = (√2 - √6)/4.
c) Evaluating tan(5π/12):
To express 5π/12 in terms of known angles, we can write it as π/4 + π/6.
Using the formula for the tangent of a sum of angles: tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B)), we can substitute A = π/4 and B = π/6 to get:
tan(5π/12) = (tan(π/4) + tan(π/6))/(1 - tan(π/4)tan(π/6))
The values of tan(π/4) and tan(π/6) are known. Substitute these values and simplify the expression:
tan(5π/12) = (1 + √3)/(1 - √3/3)
= (3 + √3)/3(1 - √3/3)
= (3 + √3)/(3 - √3)
Therefore, tan(5π/12) = (3 + √3)/(3 - √3).
d) Evaluating sin(11π/12):
Using the formula for the sine of a sum of angles: sin(A + B) = sin(A)cos(B) + cos(A)sin(B), we can express 11π/12 as 3π/4 + π/6.
sin(11π/12) = sin(3π/4 + π/6)
The values of sin(3π/4), sin(π/6), cos(3π/4), and cos(π/6) are known. Substitute these values and simplify the expression:
sin(11π/12) = (1/√2)(√2/2) + (√3/2)(√2/2)
= (√2/2 + √6/2)
= (√2 + √6)/2√2
Rationalizing the denominator by multiplying the numerator and denominator by √2:
sin(11π/12) = (√2 + √6)/4
Therefore, sin(11π/12) = (√2 + √6)/4.
Thus, we have evaluated all the given trigonometric functions using compound angle formulas.