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 θ)

We define the three trigonometrical ratios sine θ, cosine θ, and tangent θ as follows (we normally write these in the shortened forms sin θ, cos θ, and 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:

Sin θ = Opposite/Hypotenuse,

Cos θ = Adjacent/Hypotenuse, and

Tan θ = 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^-1θ 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 θ as `"opp"/"hyp"`;

cos θ as `"adj"/"hyp"`, and

tan θ as `"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 ».