A ratio is a statement of a mathematical relationship comparing two quantities, often represented as a fraction. If we consider an angle $\theta$θ in a right-angled triangle, we can construct various ratios to compare the lengths of the sides. Special relationships that exist in right-angled triangles are called trigonometric ratios.

Through investigation we can see that there is a definite relationship between the angles in a right-angled triangle and the ratio of sides. We can use these trigonometric ratios to find unknown angles and sides of a triangle.

There are 3 basic trigonometric ratios that relate sides and angles together. They have the special names of tangent, sine and cosine. The names of the relationships date back to 499AD, with ties to Latin and Sanskrit. Actually the word sine is thought to be derived from a translation gone wrong! Regardless the names for these relationships have stuck.

We often shorten the names tangent, sine and cosine to tan, sin and cos respectively.

Trigonometric ratios

$\sin\theta$sinθ = $\frac{Opposite}{Hypotenuse}$OppositeHypotenuse = $\frac{b}{c}$bc

$\cos\theta$cosθ = $\frac{Adjacent}{Hypotenuse}$AdjacentHypotenuse = $\frac{a}{c}$ac

$\tan\theta$tanθ = $\frac{Opposite}{Adjacent}$OppositeAdjacent = $\frac{b}{a}$ba

Here is a picture of the above relationships, and for some people the mnemonic of SOHCAHTOA at the bottom is helpful to remember the sides that apply to the different ratios of sine, cosine and tangent.

This applet will let you explore different trigonometric ratios, particularly which sides are needed for a particular ratio. If you would like to see this applet in action, watch this video.

Created with Geogebra

Another relationship

Let's have a look at just one more special relationship.

If we know the sine and cosine ratios for a particular angle,

$\sin\theta=\frac{Opposite}{Hypotenuse}$sinθ=OppositeHypotenuse

$\cos\theta=\frac{Adjacent}{Hypotenuse}$cosθ=AdjacentHypotenuse

Then we can construct a new relationship for $\text{sine }\div\text{cosine }$sine ÷cosine

$\frac{\sin\theta}{\cos\theta}$`sinθcosθ`	$=$=	$\frac{\left(\frac{Opposite}{Hypotenuse}\right)}{\left(\frac{Adjacent}{Hypotenuse}\right)}$(`OppositeHypotenuse`)(`AdjacentHypotenuse`)
	$=$=	$\frac{Opposite}{Hypotenuse}\times\frac{Hypotenuse}{Adjacent}$`OppositeHypotenuse`×`HypotenuseAdjacent`
	$=$=	$\frac{Opposite}{Adjacent}$`OppositeAdjacent`
	$=$=	$\tan\theta$`tanθ`

Algebraically we have just shown that $\tan\theta=\frac{\sin\theta}{\cos\theta}$tanθ=sinθcosθ

That is, the tangent ratio of an angle is the same as dividing its sine ratio by its cosine ratio.

Here are some worked examples.

Question 1

Write down the ratio represented by $\cos\theta$cosθ.

A right-angled triangle labeled with vertices A, B, and C is shown. The angle at vertex A is labeled $\alpha$`α`. The angle at vertex B is labeled $\theta$`θ`. The angle at vertex C is a right angle as indicated by a small square. Opposite vertex C is side AB, which is the hypotenuse. Side BC is adjacent to angle $\theta$`θ`. Side BC is opposite angle $\alpha$`α`. Side AC is adjacent to angle $\alpha$`α`. Side AC is opposite angle $\theta$`θ`.

Question 2

Find the value of $\tan\theta$tanθ within $\triangle ABC$△ABC.

A triangle labeled with vertices A, B, and C is shown. The angle at vertex A is labeled $\alpha$`α`. The angle at vertex B is labeled $\theta$`θ`. The angle at vertex C is a right angle as indicated by a square square. Side AB is the hypotenuse and is labeled " $60$ ," indicating its length. Side $BC$`BC` is labeled " $20$ ," indicating its length. Side BC is adjacent to angle $\theta$`θ`. Side BC is opposite angle $\alpha$`α`. Side AC is labeled " $35$ ," indicating its length. Side AC is adjacent to angle $\alpha$`α`. Side AC is opposite angle $\theta$`θ`.

Question 3

Consider the triangle in the figure. If $\sin\theta=\frac{4}{5}$sinθ=45:

A right-angled triangle is illustrated with vertices labeled $A$`A`, $B$`B`, and $C$`C`. $interval(AC)$`interval`(`AC`) is labeled with a length of $x$`x` units is adjacent to $\angle BCA$∠`BCA` of vertex $C$`C`. $AB$`AB` measures $4$4, which is opposite to $\angle BCA$∠`BCA` of vertex $C$`C`. The hypotenuse $BC$`BC` is labeled with a length of $5$5 units. A right angle is marked with a small square at vertex $A$`A` opposite to hypotenuse $BC$`BC`.

Which angle is represented by $\theta$θ?
$\angle BAC$∠BAC
A
$\angle BCA$∠BCA
B
$\angle ABC$∠ABC
C
Find the value of $\cos\theta$cosθ. Express your answer as a simplified fraction.
Find the value of $\tan\theta$tanθ. Express your answer as a simplified fraction.

Outcomes

10.T.IT.1

Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios, whichever are defined at 0° and 90°. Values (with proofs) of the trigonometric ratios of 30°, 45° and 60°. Relationships between the ratios.

Evaluate sine, cosine and tangent ratios

Trigonometric Ratios