Trigonometric Ratios
from Right Triangle Similarity

Discover how similar right triangles guarantee that \(\sin\), \(\cos\), and \(\tan\) depend only on the angle, not the triangle's size.

Geometry

What Is a Right Triangle?

Why Similarity Matters

The Three Trigonometric Ratios

\(\sin\theta\)

\(\displaystyle\frac{\text{opposite}}{\text{hypotenuse}}\)

\(\cos\theta\)

\(\displaystyle\frac{\text{adjacent}}{\text{hypotenuse}}\)

\(\tan\theta\)

\(\displaystyle\frac{\text{opposite}}{\text{adjacent}}\)

Core Theorem: Ratios Are Angle-Dependent

Because all right triangles with the same acute angle \(\theta\) are similar (by AA), corresponding side ratios are equal. This means \(\sin\theta\), \(\cos\theta\), and \(\tan\theta\) are functions of the angle alone: they do not depend on the size of the triangle.

Visualizing the Constant Ratio

No matter how large or small you draw the triangle, if angle \(\theta\) stays the same, the ratio of any two sides remains constant. That constant is the trig ratio.

\(\sin\theta\) vs \(\cos\theta\): Key Distinction

	\(\sin\theta\)	\(\cos\theta\)
Numerator	Opposite side	Adjacent side
Denominator	Hypotenuse	Hypotenuse
At \(0^\circ\)	0	1
At \(45^\circ\)	\(\frac{\sqrt{2}}{2}\)	\(\frac{\sqrt{2}}{2}\)
At \(90^\circ\)	1	0
Co-function	\(\sin\theta = \cos(90^\circ - \theta)\)	\(\cos\theta = \sin(90^\circ - \theta)\)

\(\tan\theta\) and the Ratio of Ratios

Tangent connects sine and cosine through a simple identity:

\(\tan 30^\circ = \frac{1}{\sqrt{3}}\)

Opposite is shorter than adjacent

\(\tan 45^\circ = 1\)

Opposite equals adjacent

\(\tan 60^\circ = \sqrt{3}\)

Opposite is longer than adjacent

Special Triangles: 45-45-90 and 30-60-90

How to Apply Trig Ratios

1

Identify the right angle. Confirm the triangle has a \(90^\circ\) angle.

2

Choose the reference angle \(\theta\). Pick the acute angle you know or need.

3

Label the sides. Identify opposite, adjacent, and hypotenuse relative to \(\theta\).

4

Pick the right ratio. Match the two sides you know (or want) to SOH, CAH, or TOA.

5

Set up the equation and solve. Substitute known values and solve for the unknown.

Example 1: Finding a Missing Side

Example 2: Finding an Angle

Example 3: Two-Step Problem

A 20-foot ladder leans against a wall at a \(65^\circ\) angle with the ground. How high up the wall does it reach, and how far is its base from the wall?

Why Does Similarity Guarantee It?

1

Both triangles have a right angle (\(90^\circ\)), so one pair of angles is already equal.

2

They share acute angle \(\theta\), giving a second pair of equal angles.

3

By AA Similarity, the triangles must be similar: all corresponding sides are proportional.

4

Proportional sides mean \(\frac{\text{opp}}{\text{hyp}}\) is the same in both triangles: that constant is \(\sin\theta\).

Common Mistakes

✕ Swapping Sides

Confusing opposite and adjacent relative to the chosen angle. Always label sides from the perspective of \(\theta\).

✕ Wrong Ratio

Using \(\sin\) when you need \(\cos\) (or vice versa). Match the two known/unknown sides to the correct ratio first.

✕ Degree vs Radian Mode

Calculator set to radians when the problem gives degrees. Check your mode before computing.

Summary of Trig Ratios

Angle	\(\sin\)	\(\cos\)	\(\tan\)
\(30^\circ\)	\(\frac{1}{2}\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{\sqrt{3}}\)
\(45^\circ\)	\(\frac{\sqrt{2}}{2}\)	\(\frac{\sqrt{2}}{2}\)	\(1\)
\(60^\circ\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{2}\)	\(\sqrt{3}\)

Question 1 of 5

In a right triangle, if the side opposite angle \(\theta\) is 7 and the hypotenuse is 25, what is \(\sin\theta\)?

\(\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{7}{25}\). Choice (a) flips the ratio, (c) gives \(\cos\theta\), and (d) gives \(\tan\theta\).

Question 2 of 5

Two right triangles both have an acute angle of \(40^\circ\). Triangle 1 has a hypotenuse of 5 and Triangle 2 has a hypotenuse of 15. Which statement is true?

By AA Similarity, both triangles share angles \(40^\circ\) and \(90^\circ\), making them similar. Similar triangles have equal side ratios, so \(\sin 40^\circ\) is the same in both.

Question 3 of 5

In a right triangle with \(\theta = 60^\circ\), which expression gives the adjacent side if the hypotenuse is 12?

\(\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}\), so adjacent \(= \text{hyp} \cdot \cos\theta = 12 \cdot \cos 60^\circ = 12 \cdot \frac{1}{2} = 6\). Choice (a) gives the opposite side, (c) is not a valid side formula, and (d) inverts the relationship.

Question 4 of 5

If \(\tan\theta = \frac{3}{4}\) in a right triangle, what is \(\sin\theta\)?

If \(\tan\theta = \frac{3}{4}\), the opposite is 3 and adjacent is 4. By the Pythagorean theorem, hypotenuse \(= \sqrt{3^2 + 4^2} = 5\). So \(\sin\theta = \frac{\text{opp}}{\text{hyp}} = \frac{3}{5}\).

Question 5 of 5

Which identity is a direct consequence of the co-function relationship from right triangle similarity?

In a right triangle, the side opposite \(\theta\) is the same side that is adjacent to \((90^\circ - \theta)\). So \(\sin\theta = \frac{\text{opp}}{\text{hyp}} = \cos(90^\circ - \theta)\). This co-function identity comes directly from the geometry of similar right triangles.

Trigonometric Ratiosfrom Right Triangle Similarity

What Is a Right Triangle?

Why Similarity Matters

The Three Trigonometric Ratios

Core Theorem: Ratios Are Angle-Dependent

Visualizing the Constant Ratio

\(\sin\theta\) vs \(\cos\theta\): Key Distinction

\(\tan\theta\) and the Ratio of Ratios

Special Triangles: 45-45-90 and 30-60-90

How to Apply Trig Ratios

Example 1: Finding a Missing Side

Example 2: Finding an Angle

Example 3: Two-Step Problem

Why Does Similarity Guarantee It?

Common Mistakes

Summary of Trig Ratios

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

Trigonometric Ratios
from Right Triangle Similarity