The terminal side of the angle theta in standard position is defined by 5x - 6y = 0, x ≥ 0. Find cosine theta.

Answer 1

Solution:

the equation of the line is written in slope intercept form y = mx+b

5x-6y=0. Rearrange 6y = 5x, y=(5/6)x with the slope of the line being m = 5/6 and b = 0 meaning that the y intercept is 0 meaning that the line passes through the point (x,0) substituting through the equation y=0, then x =0 and the terminal side line of the angle θ passes through the origin (0,0)

The slope of the line is tangent of the angle θ between the x axis and the line such that tanθ=5/6 with 5 being the opposite leg and 6 the adjacent leg of the right angle triangle. The hypotenuse of the triangle is the square root of √ (5²+6²) = 7.81, and cos θ = adjacent/ hypotenuse = 6/7.81=0.768 say 0.77

Answer 2

5x - 6y = 0

5x = 6y

y = (5/6)x

Graphing this, and looking at a slope triangle:

Since slope is rise/run (Δy/Δx), you can consider the cosine as 6/hypotenuse and you can solve for the hypotenuse using the Pythagorean Theorem.

Does this help?