Given cot A = 3 over 4 and that angle A is in Quadrant I, find the exact value of csc A in simplest radical form using a rational denominator

Answer 1

You do problems like this by drawing a right triangle with the given characteristics, using the Pythagorean Theorem to determine the missing side, and then finding the desired trig function.

For example, if cotA = 3/7 = Adjacent/Opposite in Quadrant 1, then draw a right triangle in Q1 with adjacent side = 3 and opposite side = 7. Then by the Pythagorean Theorem the hypotenuse is equal to √58. Once you know all 3 sides of the triangle you can calculate any of the trig functions. For example,

cosA = Adjacent/Hypotenuse = 3/√58 = 3√58/58.

Answer 2

cot A = 3/4

cot A = 1/tan A

Therefore, tan A = 4/3

By SOHCAHTOA, tan is equal to opposite/adjacent.

This means opposite = 4 and adjacent = 3. We can use these two sides to solve for the hypotenuse using Pythagorean Theorem.

(3^2) + (4^2) = c^2

9 + 16 = c^2

c^2 = 25

c = 5

Therefore, the hypotenuse is 5.

csc A = 1/sin A

sin A = opposite/hypotenuse

Using the values we have derived, sin A = 4/5

Therefore csc A, the reciprocal of sin A, is equal to 5/4

csc A = 5/4