How do I find the other 5 trigonometric functions with just a cosecant value and what quadrant it is in?

Question

I know how to find sin (it is the reciprocal of csc) but with out a cos value I cannot figure out how to find tan, cot, cos and sec

Answer 1

In order to find the other five trigonometric functions when you are given only cosecant and the quadrant, you can solve it entirely using a triangle.

If you recall, there is a mnemonic called SOH CAH TOA which helps you remember how to find the missing sides in a right triangle. What it means is:

sin(x) = opposite / hypotenuse

cos(x) = adjacent / hypotenuse

tan(x) = opposite / adjacent

From these, the reciprocal functions come from flipping the ratios. Since sine is opposite over hypotenuse, cosecant is its reciprocal, so:

csc(x) = hypotenuse / opposite

Lets understand this by doing an example:

Given:

csc(x) = 13/5 and x is in Quadrant II

Step 1:

Since csc(x) = hypotenuse / opposite:

hypotenuse = 13

opposite = 5

Step 2:

We can find the adjacent side by using the Pythagorean theorem:

a² + b² = c²

a and b can be opposite or adjacent but c is always hypotenuse so lets make a the opposite side so we can solve for b which would represent the adjacent side:

5² + b² = 13²

25 + b² = 169

b² = 169 - 25

b² = 144

b = 12

So, adjacent = 12

Step 3:

In Quadrant II, sine is positive, but cosine and tangent are negative (as well as their reciprocals).

Step 4:

Now we can find all the other trigonometric functions:

sin(x) = opposite / hypotenuse = 5/13

cos(x) = adjacent / hypotenuse = -12/13

tan(x) = opposite / adjacent = -5/12

csc(x) = 13/5 (Given)

sec(x) = hypotenuse / adjacent = -13/12

cot(x) = adjacent / opposite = -12/5

So using SOH CAH TOA, you can interpret cosecant as hypotenuse over opposite, build a triangle, find the missing side, and then use the trig ratios to get all the other trigonometric functions.

Answer 2

let csc = 17/8 = hypotenuse/opposite side

sin = opposite side/hypotenuse = 8/17

adjacent side^2 + opposite side^2 = hypotenuse^2

adjacent side^2 + 8^2 = 17^2

adjacent side^2 + 64 = 289

adjacent side^2 = 225

Take the square root of both sides

adjacent side = 15

cos = adj/hyp = 15/17

sec = hyp/adj = 17/15

tan = opp/adj = 8/15

cot = adj/opp = 15/8

Answer 3

If you have sin Φ = x/hyp, couldn't you use this fact and a²+b² = c² to solve for the missing side of the triangle made by the angle? Once you know the length of all three sides you can write all the other trig functions.

The quadrant depends on the actual angle in question.