value of expression (cos5x + cos3x)/ (sin5x - sin3x) where x = (3.14/8)
what, no calculator?
since sin 5pi/8 = sin 3pi/8, and cos 5pi/8 = -cos 3pi/8, the fraction is 0/0.
So, using l'Hospital's Rule, we get
(-5sin 5pi/8 - 3sin 3pi/8)/(5cos 5pi/8 - 3cos 3pi/8)
= -8sin(3pi/8)/-8cos(3pi/8)
= tan 3pi/8
= 2.4142
Using the approximate value given, we get 2.41557
To find the value of the expression (cos5x + cos3x) / (sin5x - sin3x) where x = (3.14/8), we can substitute the given value of x into the expression and then calculate.
Step 1: Substitute the value of x into the expression:
cos(5 * (3.14/8)) + cos(3 * (3.14/8)) / (sin(5 * (3.14/8)) - sin(3 * (3.14/8)))
Step 2: Simplify the trigonometric functions using a calculator:
cos(1.96349) + cos(1.1781) / (sin(1.96349) - sin(1.1781))
Step 3: Calculate the cosine and sine values using a calculator:
-0.49634 + 0.37082 / (0.86849 - 0.9217)
Step 4: Simplify the numerator and denominator separately:
-0.12552 / -0.05321
Step 5: Divide the numerator by the denominator:
2.362339
Therefore, the value of the expression (cos5x + cos3x) / (sin5x - sin3x), where x = (3.14/8), is approximately 2.362339.