Give that x is angle in the first quadrant such that 8sin 2x + 2cosx – 5 = 0
Find:
cosx
tanx
8sin2x + 2cosx - 5 = 0
16 sinx cosx + 2cosx - 5 = 0
16sinx cosx = 5 - 2cosx
256 sin^2x cos^2x = 25 - 20cosx + 4cos^2x
256cos^2x - 256cos^4x = 25 - 20cosx + 4cos^2x
256cos^4x - 252cos^2x - 20cosx + 25 = 0
I don't see any easy roots there, so a graphical approach may be your best bet.
To find the values of cos(x) and tan(x), we can simplify the given equation 8sin(2x) + 2cos(x) - 5 = 0 and then solve for x. Let's go step by step:
Step 1: Use the double angle formula for sine, which states that sin(2x) = 2sin(x)cos(x). Replace sin(2x) with 2sin(x)cos(x) in the equation:
8 * 2sin(x)cos(x) + 2cos(x) - 5 = 0
Step 2: Distribute 8 to both terms in the first part of the equation:
16sin(x)cos(x) + 2cos(x) - 5 = 0
Step 3: Factor out cos(x) from the equation:
cos(x)(16sin(x) + 2) - 5 = 0
Step 4: Add 5 to both sides of the equation:
cos(x)(16sin(x) + 2) = 5
Step 5: Divide both sides of the equation by (16sin(x) + 2):
cos(x) = 5 / (16sin(x) + 2)
Now, since x is an angle in the first quadrant, we know that both sin(x) and cos(x) are positive. Therefore, the value of cos(x) must also be positive.
To find the value of tan(x), we can use the identity tan(x) = sin(x) / cos(x). Since we already have the equation for cos(x), we can substitute it into the equation:
tan(x) = sin(x) / cos(x)
= sin(x) / (5 / (16sin(x) + 2))
Now, to simplify further, we can multiply the numerator and denominator of the right side by (16sin(x) + 2):
tan(x) = (sin(x) * (16sin(x) + 2)) / 5
Therefore, we have found the values of cos(x) and tan(x) in terms of sin(x).
Note: To find the actual numerical values of cos(x) and tan(x), you would need to solve the equation 8sin(2x) + 2cos(x) - 5 = 0 and then substitute the value of x into the expressions for cos(x) and tan(x) derived above.