Question

What are the six trigonometric ratios, and how are some of them related to each other(which are reciprocals of which)?

Answer 1

Step-by-step explanation:

Consider the following right triangle:

In this triangle

x = adjacent side to the angle theta.

y = opposite side to the angle theta.

h= hypotenuse.

Now, by definition, we have the following trigonometric ratios:

`\cos(\theta)=\frac{adjacent\text{ side to the angle }\theta}{hypotenuse}\text{ =}(x)/(h)`
`\sin(\theta)=\frac{opposite\text{ side to the angle }\theta}{hypotenuse}=(y)/(h)`
`tan(\theta)=\frac{opposite\text{ side to the angle }\theta}{adjacent\text{ side to the angle }\theta}=\text{ }(y)/(x)=\frac{y\text{ /h}}{x\text{ /h}}\text{ =}(\sin(\theta))/(\cos(\theta))`

and according to the above trigonometric ratio, we get:

`cotan(\theta)=(\cos(\theta))/(\sin(\theta))`

On the other hand, we get the following reciprocals:

`csc(\theta)=(1)/(\sin(\theta))`

and

`sec(\theta)=(1)/(cos(\theta))`

we can conclude that the correct answer is:

Answer:

The six trigonometric ratios:

`\cos(\theta)=\frac{adjacent\text{ side to the angle }\theta}{hypotenuse}\text{ }`

`\sin(\theta)=\frac{opposite\text{ side to the angle }\theta}{hypotenuse}`

`tan(\theta)=\frac{opposite\text{ side to the angle }\theta}{adjacent\text{ side to the angle }\theta}=\text{ }(\sin(\theta))/(\cos(\theta))`

`csc(\theta)=(1)/(\sin(\theta))`

`sec(\theta)=(1)/(cos(\theta))`

`cotan(\theta)=(\cos(\theta))/(\sin(\theta))`