The quotient properties of tangent and cotangent are: tanx = sinx ÷ cosx and cotx = cosx ÷ sinx. why are these true based on the unit circle?

Hope this helps:

Given The unit circle, a radius can be drawn from the origin to any point on the circle in the direction of theta relative to the x axis. This creates the coordinate (x,y)

Drawing a right triangle from this gives you a horizontal length of x, vertical length of y and a hypotenuse of 1.

Use SOHCAHTOA to find tan (theta) and cot (theta):

sin(theta) = y/1 = y

cos(theta) = x/1 = x

tan(theta) = O/A = y/x = sin(theta) / cos(theta)

cot(theta) = A/O = x/y = cos(theta) / sin(theta)

Answer: d. 39 cmin the diagram, the approximate length of the minor arc de is 39 centimetres.step-by-step explanation: the formula for the length of an arc in a circle is as displayed below.let the length of the minor arc de = xx = 2( pi )r × angle of sector / 360°from this, we will substitute the values given from the problem into the formula to obtain the length of the minor arc de.x = 2( pi )r × angle of sector / 360°r = radius = 25cmpi as an approximate value = 3.14angle of sector = 120°x = 2( 3.14 )( 25 ) × 120° / 360°x = 157 × 1 / 4x = 39.25x = 39 ( rounded to the nearest whole number as an approximate value )