Relationships Between Trigonometric Ratios

students, in geometry and trigonometry, one of the most useful ideas is that the three basic trigonometric ratios are connected. This lesson shows how $\sin \theta$, $\cos \theta$, and $\tan \theta$ are related, why those relationships matter, and how you can use them to solve problems in right triangles, coordinate geometry, and trigonometric equations 🌟

Introduction: the big idea

The main goal of this lesson is to understand that trigonometric ratios are not separate facts to memorize; they are linked by identities. The most important of these is the Pythagorean identity $\sin^2\theta+\cos^2\theta=1$. From this, other relationships can be built, such as $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$ and $1+\tan^2\theta=\sec^2\theta$.

These relationships help you:

move from one ratio to another,
solve exact values in triangles and on the unit circle,
simplify expressions,
prove identities,
and connect trigonometry to graphs and coordinate geometry.

A real-world example is navigation 🚗. If a ship or drone knows an angle and one trigonometric ratio, it can use these links to find other quantities like height, horizontal distance, or direction.

The three primary trigonometric ratios

In a right triangle, the trigonometric ratios are defined using the sides of the triangle. For an acute angle $\theta$:

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

These ratios describe shape, not size. That means two similar triangles with the same angle $\theta$ have the same values of $\sin\theta$, $\cos\theta$, and $\tan\theta$.

The key relationship between them comes from combining the definitions:

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

This means that tangent can be written as a ratio of sine and cosine. This is extremely useful because if you know $\sin\theta$ and $\cos\theta$, you automatically know $\tan\theta$.

For example, if $\sin\theta=\dfrac{3}{5}$ and $\cos\theta=\dfrac{4}{5}$, then

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

The Pythagorean identity and why it works

The most important relationship comes from the Pythagorean theorem. In a right triangle with hypotenuse $r$, opposite side $y$, and adjacent side $x$, the theorem says:

$ x^2+y^2=r^2$

Divide every term by $r^2$:

$\left(\frac{x}{r}\right)^2+\left(\frac{y}{r}\right)^2=1$

Using the definitions of cosine and sine, this becomes:

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

This is called the Pythagorean identity.

It is true for all angles where sine and cosine are defined, not just acute angles. On the unit circle, it is especially clear: the point on the circle has coordinates $(\cos\theta,\sin\theta)$, and because every point on the unit circle satisfies $x^2+y^2=1$, the identity follows directly.

This identity helps in many ways. If you know one of $\sin\theta$ or $\cos\theta$, you can often find the other by rearranging:

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

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

Then you take a square root, but you must choose the correct sign based on the quadrant.

Example: if $\cos\theta=\dfrac{2}{3}$ and $\theta$ is in quadrant I, then

$\sin^2\theta=1-\left(\frac{2}{3}\right)^2=1-\frac{4}{9}=\frac{5}{9}$

$\sin\theta=\frac{\sqrt{5}}{3}$

because sine is positive in quadrant I.

Secant, cosecant, and cotangent relationships

In IB Mathematics, you also meet the reciprocal ratios:

$\sec\theta=\dfrac{1}{\cos\theta}$
$\csc\theta=\dfrac{1}{\sin\theta}$
$\cot\theta=\dfrac{1}{\tan\theta}$

These are linked to the primary ratios and create more identities:

$1+\tan^2\theta=\sec^2\theta$

$1+\cot^2\theta=\csc^2\theta$

These come from the Pythagorean identity. For instance, divide $\sin^2\theta+\cos^2\theta=1$ by $\cos^2\theta$:

$\frac{\sin^2\theta}{\cos^2\theta}+1=\frac{1}{\cos^2\theta}$

which becomes

$\tan^2\theta+1=\sec^2\theta.$

These identities are useful when simplifying expressions. For example, if you see $1-\sin^2\theta$, you can replace it with $\cos^2\theta$. If you see $\dfrac{\sin\theta}{\cos\theta}$, you can replace it with $\tan\theta$. This is often the fastest path in algebraic manipulation.

Solving problems with ratios and signs

A major skill in this topic is finding missing trigonometric values from limited information. The usual approach is:

Use a known identity.
Find the missing ratio.
Check the sign using the quadrant.

Suppose $\tan\theta=-\dfrac{5}{12}$ and $\theta$ is in quadrant II. Because tangent is negative in quadrant II, this is consistent. You can think of a reference triangle with opposite $5$ and adjacent $12$, giving hypotenuse $13$ by the Pythagorean theorem. Then:

$\sin\theta=\frac{5}{13}, \qquad \cos\theta=-\frac{12}{13}$

The cosine is negative in quadrant II, while sine is positive.

This sign check is very important. A value from a square root always gives a nonnegative number unless you decide the sign based on the quadrant. Forgetting the sign is one of the most common mistakes.

Another useful strategy is to start with $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$. If the question gives a ratio such as $\sin\theta$ and asks for $\tan\theta$, you may not need to construct a triangle if the identity gives a direct route.

Identities in simplifying and proving expressions

Trigonometric identities are equations that are true for all valid angles. In IB Mathematics: Analysis and Approaches HL, you are often asked to simplify expressions or prove that two sides are equal.

For example, simplify

$\frac{1-\sin^2\theta}{\cos\theta}$

Using $1-\sin^2\theta=\cos^2\theta$, this becomes

$\frac{\cos^2\theta}{\cos\theta}=\cos\theta$

provided $\cos\theta\neq 0$.

Another example is proving

$\frac{\sec^2\theta-1}{\tan^2\theta}=1.$

Since $\sec^2\theta-1=\tan^2\theta$, the numerator becomes $\tan^2\theta$, so the whole expression is $1$.

When proving identities, a good strategy is to work on one side only and transform it until it matches the other side. Do not change both sides at once, because that makes it harder to see the logic.

Connections to graphs and the unit circle

The relationships between trigonometric ratios are also visible on graphs and the unit circle. On the unit circle, an angle $\theta$ corresponds to a point $(\cos\theta,\sin\theta)$, so the identity $\sin^2\theta+\cos^2\theta=1$ is simply the equation of the circle.

This connection explains why sine and cosine graphs have the same shape but are shifted. Tangent behaves differently because it is defined by

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

so it is undefined when $\cos\theta=0$. That is why the tangent graph has vertical asymptotes.

A practical example is a Ferris wheel 🎡. If the position of a seat is modeled using coordinates, the horizontal movement can be described by $\cos\theta$ and the vertical movement by $\sin\theta$. The relationship between them keeps the seat on a circle of fixed radius.

Conclusion

students, the relationships between trigonometric ratios are powerful because they turn separate formulas into one connected system. The key ideas are $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$ and $\sin^2\theta+\cos^2\theta=1$, along with the reciprocal identities $1+\tan^2\theta=\sec^2\theta$ and $1+\cot^2\theta=\csc^2\theta$. These identities help you calculate unknown values, simplify expressions, and prove results in geometry and trigonometry.

This topic fits into the broader IB course because it supports right-triangle trigonometry, circular measure, graphing trigonometric functions, and solving equations. Mastering these links gives you a strong base for more advanced HL problem-solving.

Study Notes

$\sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}$.
The fundamental identity is $\sin^2\theta+\cos^2\theta=1$.
A very important relationship is $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$.
Reciprocal ratios are $\sec\theta=\dfrac{1}{\cos\theta}$, $\csc\theta=\dfrac{1}{\sin\theta}$, and $\cot\theta=\dfrac{1}{\tan\theta}$.
Useful derived identities are $1+\tan^2\theta=\sec^2\theta$ and $1+\cot^2\theta=\csc^2\theta$.
To find a missing ratio, use an identity and then check the sign from the quadrant.
On the unit circle, the point is $(\cos\theta,\sin\theta)$.
In proofs, usually start with one side and transform it step by step.
These relationships are essential for simplifying expressions, solving equations, and connecting trigonometry to geometry and graphs.