Consider △dfe. what are the inputs or outputs of the following trigonometric ratios? express the ratios in simplest terms. = 4/5 cos(f) = ? tan(d) = ?

sin F = 4/5
cos F = 3/5
tan D = 3/4

Explanation:
First, we have sin an angle = 4/5
In a right-angled triangle:
sin theta = opposite / hypotenuse
This means that in triangle DFE:
The hypotenuse = 5 units
one of the sides = 4 units
Use Pythagoren theorem to get the other side as follows:
hypotenuse^2 = side1^2 + side2^2
5^2 = 4^2 + side2^2
side2^2 = 9
The second side of the triangle = 3 units

The special trig functions are as follows:
sin theta = opposite / hypotenuse
cos theta = adjacent / hypotenuse
tan theta = opposite / adjacent

From the attached figure and based on the calculations above:
sin F = 4/5
cos F = 3/5
tan D = 3/4

Hope this helps :)

Intersecting secant-tangent theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.so to break it down: 1. pq acts as the tangent and ps as the secant with an intersection at p.sr = 21rp = 3x + 3pq = 4x + 4following to equation: tangent squared = the two segments multiplied by each otherso (4x + 4) ^ 2 = (3x + 3) • (21) simplifies to 16x^2 - 31x - 47 = 0 or (16x - 47)(x + 1) x = -1 or x = 47/162. pq acts as the tangent and ps as the secant with an intersection at p (again! )sr = 16rp = xpq = 15(15) ^ 2 = 16 • x225 = 16xx = 225/163. the interesting chords are proportional to each other so a ratio is possible to set up: ap dp 10 3 + x = = —— = pc pb 8 xcross multiple to get 10x = (3 + x)(8)x = 12