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 first need to solve the given equation: 8sin(2x) + 2cos(x) - 5 = 0.

Let's simplify this equation step by step:

1. We know that sin(2x) = 2sin(x)cos(x). Substituting this formula in the first term, we get: 8(2sin(x)cos(x)) + 2cos(x) - 5 = 0.

2. Multiplying out the first term, we have: 16sin(x)cos(x) + 2cos(x) - 5 = 0.

3. Factoring out the common factor of cos(x), we get: cos(x)(16sin(x) + 2) - 5 = 0.

4. Simplifying further, we have: cos(x)(16sin(x) + 2) = 5.

Now, we have two cases to consider:

Case 1: cos(x) = 0.
When cos(x) = 0, it means that x = π/2. However, we are given that x is in the first quadrant. In the first quadrant, the cosine function is positive, so this case is not valid.

Case 2: 16sin(x) + 2 = 5.
Solving this equation for sin(x), we have: 16sin(x) = 3.

Dividing both sides by 16, we get: sin(x) = 3/16.

Using the Pythagorean identity: sin^2(x) + cos^2(x) = 1, we can find cos(x).

1. Substitute sin(x) = 3/16 into the Pythagorean identity: (3/16)^2 + cos^2(x) = 1.

2. Solving for cos^2(x), we have: 1 - (9/256) = cos^2(x).

3. Taking the square root of both sides, we get: cos(x) = √(256/256 - 9/256) = √(247/256) = ± √(247)/16.

Since x is in the first quadrant, we can ignore the negative value. Therefore, cos(x) = √(247)/16.

To find tan(x), we can use the identity: tan(x) = sin(x)/cos(x).

1. Substitute sin(x) = 3/16 and cos(x) = √(247)/16 into the formula: tan(x) = (3/16)/(√(247)/16).

2. Simplifying, we get: tan(x) = 3/√(247).

Therefore, the values of cos(x) and tan(x) are: cos(x) = √(247)/16 and tan(x) = 3/√(247).