Relationship Between Trigonometric Ratios

students, imagine standing on a beach and looking at a lighthouse 🌊. If you know how far you are from it and the angle you’re looking up, you can find its height. That idea is the heart of trigonometric ratios: they connect angles and side lengths in right-angled triangles. In IB Mathematics Analysis and Approaches SL, this relationship helps you solve geometry problems, model real situations, and build a stronger understanding of trigonometric functions.

What trigonometric ratios mean

The three main trigonometric ratios are sine, cosine, and tangent. They are defined for a right-angled triangle using an angle and the side lengths relative to that angle.

For an angle $\theta$ in a right-angled triangle:

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$
$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
$\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

The words “opposite” and “adjacent” depend on the angle you choose. The hypotenuse is always the longest side and lies opposite the right angle.

These ratios are called “ratios” because they compare side lengths. They are not fixed lengths. Instead, they depend on the angle. That means every angle in a right triangle has a matching set of trig ratios 📐.

Why these ratios matter

The relationship between trigonometric ratios and angles lets us solve triangles. If you know one acute angle and one side, you can often find the missing sides. If you know two sides, you can find an angle. This is useful in surveying, navigation, architecture, and physics.

For example, suppose a tree casts a shadow of $12\,\text{m}$ and the angle of elevation from the tip of the shadow to the top of the tree is $35^\circ$. If $h$ is the height of the tree, then

$\tan 35^\circ = \frac{h}{12}$

$h = 12\tan 35^\circ$

This gives a height estimate using a trig ratio.

The key relationship between the ratios

The most important link between sine, cosine, and tangent is

$\tan\theta = \frac{\sin\theta}{\cos\theta}$

This formula shows that tangent is not independent. It can be built from sine and cosine. It also explains why $\tan\theta$ is undefined when $\cos\theta = 0$, because division by zero is not allowed.

Another major relationship comes from the Pythagorean theorem. In a right triangle, if the hypotenuse is $1$, then the side lengths corresponding to $\sin\theta$ and $\cos\theta$ satisfy

$\sin^2\theta + \cos^2\theta = 1$

This identity is true for every angle $\theta$ where the trig functions are defined. It is one of the most important formulas in trigonometry.

How the identity is formed

Think of a right triangle with hypotenuse $1$. If the side opposite $\theta$ is $\sin\theta$ and the side adjacent to $\theta$ is $\cos\theta$, then the Pythagorean theorem gives

$(\sin\theta)^2 + (\cos\theta)^2 = 1^2$

which becomes

$\sin^2\theta + \cos^2\theta = 1$

This identity helps simplify expressions and solve equations. For example, if you know $\sin\theta = \frac{3}{5}$ and $\theta$ is acute, then

$\cos$$\theta$ = $\sqrt{1-\sin^2\theta}$ = $\sqrt{1-\frac{9}{25}}$ = $\frac{4}{5}$

and then

\tan$\theta$ = $\frac{\sin\theta}{\cos\theta}$ = $\frac{3/5}{4/5}$ = $\frac{3}{4}$

Using ratios to solve problems

A big part of IB trigonometry is deciding which ratio to use. A simple memory aid is SOHCAHTOA:

$\sin = \frac{\text{opposite}}{\text{hypotenuse}}$
$\cos = \frac{\text{adjacent}}{\text{hypotenuse}}$
$\tan = \frac{\text{opposite}}{\text{adjacent}}$

Suppose students is asked to find the height of a flagpole. If the angle of elevation from a point on the ground is $48^\circ$ and the point is $20\,\text{m}$ from the base, then $h$ satisfies

$\tan 48^\circ = \frac{h}{20}$

$h = 20\tan 48^\circ$

If the question instead gave the hypotenuse, then sine or cosine might be better. For example, if a ladder of length $7\,\text{m}$ leans against a wall and makes an angle of $65^\circ$ with the ground, the height reached is

$7\sin 65^\circ$

because the height is opposite the angle and the ladder is the hypotenuse.

Choosing the correct ratio

To choose the right trig ratio, follow these steps:

Identify the angle.
Mark the known and unknown sides.
Decide which ratio includes those sides.
Substitute values and solve.

This careful process reduces mistakes. In exams, students often lose marks by using the wrong side names relative to the chosen angle.

Relationship with angles and the unit circle

Although the basic definitions come from right triangles, the trig ratios are also connected to the unit circle. In the unit circle, a point on the circle has coordinates

$(\cos\theta,\sin\theta)$

for an angle $\theta$ measured from the positive $x$-axis. This gives a deeper meaning to the relationship between sine and cosine: they are the $y$- and $x$-coordinates of a point on a circle of radius $1$.

Because the radius is $1$, the Pythagorean identity follows immediately:

$(\cos\theta)^2 + (\sin\theta)^2 = 1$

The unit circle also explains why trig ratios can be positive or negative depending on the quadrant. For example, in quadrant II, $\sin\theta$ is positive but $\cos\theta$ is negative, so

$\tan\theta = \frac{\sin\theta}{\cos\theta}$

is negative.

This is important for solving trig equations and understanding graphs later in the course.

Example with angle signs

If $\theta$ is in quadrant IV and $\cos\theta = \frac{5}{13}$, then $\sin\theta$ must be negative. Using

$\sin^2\theta + \cos^2\theta = 1$

we get

$\sin^2$$\theta$ = 1-$\left($$\frac{5}{13}$$\right)^2$ = 1-$\frac{25}{169}$ = $\frac{144}{169}$

$\sin\theta = -\frac{12}{13}$

because quadrant IV has negative sine values. Then

$\tan\theta = \frac{-12/13}{5/13} = -\frac{12}{5}$

Trigonometric identities and equation solving

The relationship between trigonometric ratios becomes especially powerful when solving equations. A trig equation is an equation containing trig expressions, such as

$2\sin\theta = 1$

$\sin^2\theta + \cos^2\theta = 1$

Suppose you need to solve

$2\sin\theta = 1$

Then

$\sin\theta = \frac{1}{2}$

On $0^\circ \leq \theta < 360^\circ$, this gives

$\theta = 30^\circ, 150^\circ$

because sine is positive in quadrants I and II.

Now consider

$\tan\theta = \sqrt{3}$

The reference angle is $60^\circ$, and tangent is positive in quadrants I and III, so

$\theta = 60^\circ, 240^\circ$

These solutions show that the relationship between ratios is not just about triangles. It also helps find all angles that satisfy an equation.

Using identities to simplify

Sometimes expressions look complicated, but identities make them easier. For example,

$\frac{\sin\theta}{\cos\theta}$

can be rewritten as

$\tan\theta$

And if you see $1-\sin^2\theta$, you can replace it with

$\cos^2\theta$

This kind of rewriting is very common in IB problems. It helps reduce complicated expressions to simpler ones.

Conclusion

The relationship between trigonometric ratios is one of the foundation ideas of Geometry and Trigonometry. The ratios $\sin\theta$, $\cos\theta$, and $\tan\theta$ connect angles to side lengths in right triangles, and they are linked by identities such as

$\tan\theta = \frac{\sin\theta}{\cos\theta}$

and

$\sin^2\theta + \cos^2\theta = 1$

These relationships help you solve triangles, interpret real-world situations, and solve trigonometric equations. They also connect directly to the unit circle, which extends trigonometry beyond right triangles. students, once you understand how the ratios work together, many later ideas in IB Mathematics become much easier to see and use ✨.

Study Notes

$\sin\theta$, $\cos\theta$, and $\tan\theta$ are ratios of side lengths in a right-angled triangle.
$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$.
$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$.
$\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$.
The identity $\sin^2\theta + \cos^2\theta = 1$ comes from the Pythagorean theorem.
The identity $\tan\theta = \frac{\sin\theta}{\cos\theta}$ shows tangent depends on sine and cosine.
If $\cos\theta = 0$, then $\tan\theta$ is undefined.
In the unit circle, a point has coordinates $\left(\cos\theta,\sin\theta\right)$.
The signs of trig ratios depend on the quadrant.
Trig ratios are used to find unknown sides, unknown angles, and solutions to trig equations.
Real-world applications include height, distance, navigation, and surveying 📏.