Give that x is angle in the first quadrant such that 8sin 2x + 2cosx – 5 = 0
Find:
cosx
tanx
To find the values of cos(x) and tan(x), we can start by rearranging the equation and using trigonometric identities.
Given: 8sin(2x) + 2cos(x) - 5 = 0
We can rewrite sin(2x) using the double angle identity: sin(2x) = 2sin(x)cos(x).
Substituting this in the equation, we have: 8(2sin(x)cos(x)) + 2cos(x) - 5 = 0
Rearranging the terms, we get: 16sin(x)cos(x) + 2cos(x) - 5 = 0
Factor out cos(x) from the first two terms: cos(x)(16sin(x) + 2) - 5 = 0
Now, we can solve for cos(x) by isolating the factor cos(x):
cos(x)(16sin(x) + 2) = 5
Divide both sides by (16sin(x) + 2): cos(x) = 5 / (16sin(x) + 2)
Since x is an angle in the first quadrant, sin(x) and cos(x) are both positive. Therefore, we can simplify the equation:
cos(x) = 5 / (16sin(x) + 2) (positive value in the first quadrant)
Next, let's find tan(x) using the known relationship: tan(x) = sin(x) / cos(x).
Substituting the value of cos(x) we found above:
tan(x) = sin(x) / (5 / (16sin(x) + 2))
Simplifying further:
tan(x) = (16sin(x) + 2) / 5
So, the values of cos(x) and tan(x) in terms of sin(x) are:
cos(x) = 5 / (16sin(x) + 2)
tan(x) = (16sin(x) + 2) / 5