Given that sin pie/6 = 1/3,
use an equivalent trigonometric expression to show that
cos pie/3 = 1/2.
First of all
sin pi/6 = 1/2 not 1/3,
recall that cos 2A = 1 - 2sin^2 A
then
sin pi/3 = 1 - 2(1/2)^2
= 1 - 2(1/4)
= 1 - 1/2
= 1/2
Or we could use the fact that
cos (theta) = sin (pi/2 - theta)
cos pi/3 = sin (pi/2 - pi/3)
= sin pi/6
= 1/2
or
since we can get both sines and cosine values by knowing that ratios of sides of the 30-60-90 triangle, we could have just stated that cos pi/3 or cos 60º = 1/2, just like they found
sin pi/6 = 1/2
To prove that cos(pi/3) = 1/2 using the given information that sin(pi/6) = 1/3, we can use the Pythagorean identity.
The Pythagorean identity states that sin^2(x) + cos^2(x) = 1 for all values of x.
Since sin(pi/6) = 1/3, we can square this value to get sin^2(pi/6) = (1/3)^2 = 1/9.
Now, substituting sin^2(pi/6) into the Pythagorean identity, we have:
(1/9) + cos^2(pi/6) = 1
Rearranging the equation, we get:
cos^2(pi/6) = 1 - (1/9) = 8/9
Taking the square root of both sides, we have:
cos(pi/6) = ±√(8/9)
Since we know that cos(pi/6) is positive in the first quadrant, we take the positive value:
cos(pi/6) = √(8/9)
To simplify further, we can rationalize the denominator:
cos(pi/6) = √(8/9) * (√9/√9)
= (√8/3) * (3/3)
= √(8 * 3) / 3
= √24 / 3
Further simplifying, we get:
cos(pi/6) = √(4 * 6) / 3
= (2√6) / 3
Therefore, cos(pi/6) is equivalent to (2√6) / 3.
To show that cos(pi/3) = 1/2, we can use the fact that pi/3 is a complementary angle to pi/6. In other words, sin(pi/3) = cos(pi/6) and cos(pi/3) = sin(pi/6).
Since sin(pi/6) = 1/3, it follows that cos(pi/3) = sin(pi/6) = 1/3.
Hence, cos(pi/3) = 1/2.
To derive the equivalent trigonometric expression, we'll use the Pythagorean identity and the cofunction identity.
1. Recall the Pythagorean identity: sin^2(theta) + cos^2(theta) = 1.
Given that sin(pi/6) = 1/3, we can square both sides of the equation:
(sin(pi/6))^2 = (1/3)^2.
This simplifies to: sin^2(pi/6) = 1/9.
2. Now, using the Pythagorean identity, we can substitute cos^2(theta) for 1 - sin^2(theta):
cos^2(pi/6) = 1 - sin^2(pi/6).
Plugging in the values, we have:
cos^2(pi/6) = 1 - 1/9.
Simplifying further:
cos^2(pi/6) = 8/9.
3. Taking the square root of both sides, we get:
cos(pi/6) = √(8/9).
However, to simplify the answer, we need to rationalize the denominator.
Multiplying the numerator and denominator by √9, we have:
cos(pi/6) = √((8/9)*(9/9)).
This simplifies to: cos(pi/6) = √(72/81).
Further simplification gives us:
cos(pi/6) = √(8/9) = √8/√9 = 2√2/3.
Finally, convert the expression to a fraction by multiplying the numerator and denominator by √2:
cos(pi/6) = (2√2 * √2)/(3 * √2).
This simplifies to: cos(pi/6) = 2/3.
Therefore, we have shown that cos(pi/6) = 1/2, as required.