# 2. Sine, Cosine, Tangent and the Reciprocal Ratios

by M. Bourne

For the angle *θ* in a right-angled triangle as shown, we name the sides as:

**hypotenuse**(the side opposite the right angle)**adjacent**(the side "next to"*θ*)**opposite**(the side furthest from the angle)

We **define** the three trigonometrical ratios **sine θ**,

**cosine**, and

*θ***tangent**as follows (we normally write these in the shortened forms

*θ***sin**,

*θ***cos**, and

*θ***tan**):

*θ*`sin \ theta=text(opposite)/text(hypotenuse)` `cos \ theta =text(adjacent)/text(hypotenuse)` `tan \ theta =text(opposite)/text(adjacent)`

To remember these, many people use SOH CAH TOA, that is:

Sinθ=Opposite/Hypotenuse,

Cosθ=Adjacent/Hypotenuse, and

Tanθ=Opposite/Adjacent

### The Reciprocal Trigonometric Ratios

Often it is useful to use the reciprocal ratios, depending on the problem. (In plain English, the reciprocal of a fraction is found by turning the fraction upside down.)

`"cosecant"\ θ` is the reciprocal of `"sine"\ θ`,

`"secant"\ θ` is the reciprocal of `"cosine"\ θ`, and

`"cotangent"\ θ` is the reciprocal of `"tangent"\ θ`

We usually write these in short form as `csc\ θ`, `sec\ θ` and `cot\ θ`**.** (In some textbooks, "**csc**" is written as "**cosec**". It's the same thing.)

`csc \ theta =text(hypotenuse)/text(opposite)` `sec\ theta =text(hypotenuse)/text(adjacent)` `cot \ theta =text(adjacent)/text(opposite)`

**Important note:** There is a big difference between csc *θ* and sin^{-1}* θ*.

- The first one is a reciprocal: `csc\ theta=1/(sin\ theta)`.
- The second one involves finding an
**angle**whose sine is*θ*.

So on your calculator, don't use your sin^{-1} button to find csc *θ*.

We will meet the idea of sin^{-1}*θ* in the next section, Values of Trigonometric Functions.

## The Trigonometric Functions on the *x-y* Plane

For an angle in standard position, we define the trigonometric ratios in terms of *x*,* y* and *r*:

`sin \ theta =y/r` `cos\ theta =x/r` `tan\ theta =y/x`

Notice that we are still defining

sin

θas `"opp"/"hyp"`; cosθas `"adj"/"hyp"`, and tanθas `"opp"/"adj"`,

but we are using the specific *x*-, *y*- and *r*-values defined by the point (*x*, *y*) that the terminal side passes through. We can choose any point on that line, of course, to define our ratios.

To find *r*, we use Pythagoras' Theorem, since we have a right angled triangle:

`r=sqrt(x^2+y^2)`

Not surprisingly, the reciprocal ratios are defined similarly in terms of the *x*-, *y*- and *r*-values as follows:

`csc\ theta =r/y` `sec\ theta =r/x` `cot\ theta =x/y`

We will see some examples of finding exact values in the next section, Values of Trigonometric Functions ».

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