8.01 Trigonometric functions as right triangle ratios

Trigonometric ratios

Trigonometric ratios describe the relationships between the different sides of a right triangle. These sides are described in terms of where they are compared to one of the two acute angles.

So a trigonometric ratio tells us which two sides are in a given ratio, as well as the angle the ratio is being measured from.

In the triangle below we’ve chosen one of the acute angles, labeled $x$x. The sides of the triangle can then be referred to as the side opposite $x$x, the side adjacent to $x$x and the hypotenuse.

 

Three trigonometric ratios are:

The sine ratio: $\sin\left(x\right)=\frac{\text{Opposite}}{\text{Hypotenuse}}$sin(x)=OppositeHypotenuse​

The cosine ratio: $\cos\left(x\right)=\frac{\text{Adjacent}}{\text{Hypotenuse}}$cos(x)=AdjacentHypotenuse​

The tangent ratio: $\tan\left(x\right)=\frac{\text{Opposite}}{\text{Adjacent}}$tan(x)=OppositeAdjacent​

There are also what we call the reciprocal trigonometric ratios, each of which are a reciprocal of one of the three ratios above.

The cosecant ratio: $\csc\left(x\right)=\frac{1}{\sin\left(x\right)}=\frac{\text{Hypotenuse}}{\text{Opposite}}$csc(x)=1sin(x)​=HypotenuseOpposite​

The secant ratio: $\sec\left(x\right)=\frac{1}{\cos\left(x\right)}=\frac{\text{Hypotenuse}}{\text{Adjacent}}$sec(x)=1cos(x)​=HypotenuseAdjacent​

The cotangent ratio: $\cot\left(x\right)=\frac{1}{\tan\left(x\right)}=\frac{\text{Adjacent}}{\text{Opposite}}$cot(x)=1tan(x)​=AdjacentOpposite​

Notice similar triangles will have the same the trigonometric ratio of a particular angle, because enlarging or shrinking a triangle doesn't change the size of its angles or the ratio of its sides. 
For example, if $\tan x=\frac{2}{1}$tanx=21​ we can tell that the side opposite to $x$x is twice as big as the side adjacent to $x$x. The sides might have lengths $6$6 and $3$3, not necessarily $2$2 and $1$1.

Even though we don’t know the exact length of the sides just from knowing these ratios, we can still use one ratio to calculate another ratio.

 

Using one ratio to find another

Given one trigonometric ratio, we can construct a right triangle that has an angle and sides that match the information described by the ratio.

We can then use the Pythagorean Theorem to find the length of the other side.

Once we have all three side lengths, we can find any other trigonometric ratio of the acute angle by using the appropriate formula.

 

Worked example

Question 1

A right triangle has $\sin\theta=\frac{13}{85}$sinθ=1385​, find the value of $\csc\theta$cscθ and $\sec\theta$secθ.

Think: If $\sin\theta=\frac{13}{85}$sinθ=1385​, then this means we can make the length of the hypotenuse is $85$85 and the length of the opposite side $13$13. We need to find the length of the adjacent side for $\sec\theta$secθ.

Do: 

Finding $\csc\theta$cscθ:

$\csc\theta$cscθ	$=$=	$\frac{1}{\sin\theta}$1sinθ​
	$=$=	$\frac{1}{\frac{13}{85}}$11385​​
	$=$=	$\frac{85}{13}$8513​

Finding $\sec\theta$secθ, by first finding the adjacent side:

$a^2+b^2$a2+b2	$=$=	$c^2$c2	State the Pythagorean Theorem
$a^2+13^2$a2+132	$=$=	$85^2$852	Fill in the given information
$a^2+169$a2+169	$=$=	$7225$7225	Evaluate the squares
$a^2$a2	$=$=	$7056$7056	Subtract $169$169 from both sides
$a$a	$=$=	$\sqrt{7056}$√7056	Take the square root of both sides
$a$a	$=$=	$84$84	Simplify the square root

 

We get the triangle below

Now to find $\sec\theta$secθ

$\sec\theta$secθ	$=$=	$\frac{1}{\cos\theta}$1cosθ​
	$=$=	$\frac{\text{hypotenuse }}{\text{adjacent }}$hypotenuse adjacent ​
	$=$=	$\frac{85}{84}$8584​

 

Practice questions

Question 2

Question 3

Question 4

Question 5