find exact value cos(13pi/15)cos(-pi/5)-sin(13pi/15)sin(-pi/5)
do you see the pattern
cos(A + B) = cosAcosB - sinAsinB here ?
also remember that cos(-A) = cos (A)
give it a try
To find the exact value of cos(13π/15)cos(-π/5) - sin(13π/15)sin(-π/5), we can use the trigonometric identities for the cosine and sine of the difference of two angles.
The identity for cosine of the difference of two angles is:
cos(a - b) = cos(a)cos(b) + sin(a)sin(b)
The identity for sine of the difference of two angles is:
sin(a - b) = sin(a)cos(b) - cos(a)sin(b)
In this case, a = 13π/15 and b = -π/5. So, we have:
cos(13π/15)cos(-π/5) - sin(13π/15)sin(-π/5) = cos(13π/15 - (-π/5))
Using the identity for cosine of the difference of two angles, we can rewrite the expression as:
cos(13π/15 - (-π/5)) = cos(13π/15 + π/5)
Now, we need to simplify the expression inside the cosine function:
13π/15 + π/5 = (13π + 3π)/15 = 16π/15
Finally, we have:
cos(16π/15)
To find the exact value of cos(16π/15), we can use the unit circle or refer to trigonometric values for common angles. On the unit circle, we see that 16π/15 corresponds to an angle π/15 beyond 180°. In the second quadrant, the cosine function is negative.
Therefore, cos(16π/15) = -cos(π/15)
To find the value of cos(π/15) exactly, we can use trigonometric tables, a calculator, or trigonometric relationships like the half-angle formula, but it will result in an approximate value.
Alternatively, we can rationalize the denominator in the half-angle formula to express cos(π/15) as a square root. However, this involves more complex calculations.