find csc(arctan 8/15)?

Answer 1

let x = atan(8/15)
tanx = 8/15

RTF: 1/sinx
sec^2 x = 1 + tan^2 x
= 1 + 64 / 225 = 289 / 225
cos^2 x = 225 / 289
sin^2 x = 1- cos^2 x = 64 / 289
sinx = 8/17
cscx = 17/8

Answer 2

Let x = arctan 8/15, so tan x = 8/15. We need csc x.
Use the identity: sin^2 x + cos^2 x = 1
Dividing by sin^2 x gives:
1 + cot^2 x = csc^2 x where cot x = cos x/sin x, csc x = 1/sin x
tan x = 8/15, so cot x = 15/8, since cot x = 1/tan x.
csc^2 x = 1 + (15^2)/(8^2) = 1+225/64 = 289/64
So csc x = sqrt(289/64) = 17/8.

Answer 3

Let y = csc(arctan 8/15)... then
arccsc y = arctan 8/15.

This means csc t = y and tan t = 8/15 for some angle t. So...
tan t = 8/15
→ sin t / cos t = 8/15
→ sin²t / cos²t = 64/225
→ sin²t / (1 - sin²t) = 64/225

Solve for sin²t. Then y = 1/sint.

Answer 4

arctan 8/15 is an angle in the first quadrant. If you construct a right triangle with the angle at the origin, the hypotenuse of the right triangle equals 17 (by Pythagorean). Therefore, the cosecant of this angle is hypotenuse / opposite leg = 17/8

Answer 5

Let x = tan^(-1) (8/15)
8 and 15 are two sides at right angles in a triangle.
Third side = 17 (Hypotenuse)
cosec x = 1 / sin x
cosec x = 1 / 8 /17
cosec x = 17 / 8

Answer 6

csc(arctan(8/15)

Before we pursue this question, remember the right angle trig rules:

sin(t) = opp/hyp, cos(t) = adj/hyp, tan(t) = opp/adj

It follows that their reciprocals are:

csc(t) = hyp/opp, sec(t) = hyp/adj, cot(t) = adj/opp

Now onto the question.

Let t = arctan(8/15). Then
tan(t) = 8/15.
But remember, tan(t) = opp/adj. Let's call our angle t. Thus,

tan(t) = opp/adj = 8/15
opp = 8
adj = 15, so by Pythagoras, it follows that
hyp = sqrt(8^2 + 15^2) = sqrt(64 + 225) = sqrt(289) = 17

Therefore,

csc(t) = csc(arctan(8/15)) = hyp/opp = 17/8

Answer 7

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