Find the exact value:
cos(pi/16)cos(3pi/16)-sin(pi/16)sin(3pi/16)
4pi/16 = 1/root2
To find the exact value of the given expression, we can use the trigonometric identity for the cosine of a difference of angles:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
Let's substitute A = pi/16 and B = 3pi/16 into this identity:
cos(pi/16 - 3pi/16) = cos(pi/16)cos(3pi/16) + sin(pi/16)sin(3pi/16)
Simplifying the left side:
cos(-2pi/16) = cos(-pi/8) = cos(pi/8)
Now we have:
cos(pi/8) = cos(45°)
To find the exact value of cos(45°), we can use the special triangle or unit circle.
Method 1: Using a Special Triangle
Draw a right-angled triangle with angles 45°, 45°, and 90°. The side lengths are in a ratio of 1:1:√2.
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/ |
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/ _|_
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√2
Now we can see that the adjacent side length (the side adjacent to the angle of 45°) is 1, and the hypotenuse is √2.
By definition, cosine is equal to the adjacent side divided by the hypotenuse:
cos(45°) = 1/√2
Rationalizing the denominator, we multiply both the numerator and denominator by √2:
1/√2 * (√2/√2) = √2/2
So, cos(45°) = √2/2.
Method 2: Using the Unit Circle
On the unit circle, the angle of 45° corresponds to the point (cos(45°), sin(45°)) = (cos(π/4), sin(π/4)) = (1/√2, 1/√2).
Therefore, cos(45°) = 1/√2.
Now, substituting back this value into our original expression:
cos(pi/16)cos(3pi/16) - sin(pi/16)sin(3pi/16)
= (1/√2)(1/√2) - (1/√2)(1/√2)
= 1/2 - 1/2
= 0
Hence, the exact value of the given expression is 0.