Trigonometry Guide
Learn the side-length patterns for \(45^\circ\text{-}45^\circ\text{-}90^\circ\) and \(30^\circ\text{-}60^\circ\text{-}90^\circ\) triangles, then use them to find missing sides and exact trigonometric values.
Special right triangles have side lengths that always follow the same patterns. Once you recognize the triangle, you can find missing sides without using a calculator.
There are two important families: the \(45^\circ\text{-}45^\circ\text{-}90^\circ\) triangle and the \(30^\circ\text{-}60^\circ\text{-}90^\circ\) triangle. These relationships also explain many of the exact values found on the unit circle.
The Big Idea
Every right triangle contains one \(90^\circ\) angle, but most right triangles can have many different combinations of side lengths. Special right triangles are different because their angle measures force their sides to follow a fixed ratio.
This means that once you know one side, the other two sides can be found by multiplying or dividing by the appropriate part of the ratio.
Isosceles Right Triangle
The two legs are equal. The hypotenuse is a leg multiplied by \(\sqrt{2}\).
Explore the 45-45-90 triangleHalf of an Equilateral Triangle
The short leg is opposite \(30^\circ\), and the long leg is opposite \(60^\circ\).
Explore the 30-60-90 triangleTriangle Family 1
A \(45^\circ\text{-}45^\circ\text{-}90^\circ\) triangle is an isosceles right triangle. Because the two acute angles are equal, the two legs opposite those angles must also be equal.
If each leg has length \(x\), the hypotenuse has length \(x\sqrt{2}\).
Visual Model
First Leg
Opposite one \(45^\circ\) angle
Second Leg
Opposite the other \(45^\circ\) angle
Hypotenuse
Opposite the \(90^\circ\) angle
45-45-90 Rule
Worked Example 1
A \(45^\circ\text{-}45^\circ \text{-}90^\circ\) triangle has a leg of length \(7\). Find the hypotenuse.
The angles are \(45^\circ\), \(45^\circ\), and \(90^\circ\).
Worked Example 2
The hypotenuse of a \(45^\circ\text{-}45^\circ \text{-}90^\circ\) triangle is \(12\). Find the length of each leg.
Triangle Family 2
A \(30^\circ\text{-}60^\circ \text{-}90^\circ\) triangle can be created by cutting an equilateral triangle in half. Its three side lengths always follow the ratio \(1:\sqrt{3}:2\).
Unlike a \(45^\circ\text{-}45^\circ \text{-}90^\circ\) triangle, the two legs are not equal. You must identify the short leg and long leg by looking at the angles opposite them.
Visual Model
Short Leg
Opposite the \(30^\circ\) angle
Long Leg
Opposite the \(60^\circ\) angle
Hypotenuse
Opposite the \(90^\circ\) angle
30-60-90 Rule
Worked Example 3
A \(30^\circ\text{-}60^\circ \text{-}90^\circ\) triangle has a short leg of length \(5\). Find the long leg and the hypotenuse.
The side opposite \(30^\circ\) is the short leg, so \(x=5\).
Worked Example 4
The long leg of a \(30^\circ\text{-}60^\circ \text{-}90^\circ\) triangle is \(9\sqrt{3}\). Find the short leg and hypotenuse.
Worked Example 5
The hypotenuse of a \(30^\circ\text{-}60^\circ \text{-}90^\circ\) triangle is \(14\). Find both legs.
A Reliable Process
Do not begin by multiplying numbers immediately. First identify the triangle family and determine which side length you have been given.
Decide whether the triangle is \(45\text{-}45\text{-}90\) or \(30\text{-}60\text{-}90\).
Mark the legs and hypotenuse. For a \(30\text{-}60\text{-}90\) triangle, also identify the short and long legs.
Write the appropriate side ratio before substituting the known length.
Solve for \(x\), simplify radicals, and include units when the problem provides them.
Decision Guide
| Triangle | Known Side | Side to Find | Operation |
|---|---|---|---|
| \(45\text{-}45 \text{-}90\) | Leg | Hypotenuse | Multiply by \(\sqrt{2}\) |
| \(45\text{-}45 \text{-}90\) | Hypotenuse | Leg | Divide by \(\sqrt{2}\) |
| \(30\text{-}60 \text{-}90\) | Short leg | Long leg | Multiply by \(\sqrt{3}\) |
| \(30\text{-}60 \text{-}90\) | Short leg | Hypotenuse | Multiply by \(2\) |
| \(30\text{-}60 \text{-}90\) | Hypotenuse | Short leg | Divide by \(2\) |
| \(30\text{-}60 \text{-}90\) | Long leg | Short leg | Divide by \(\sqrt{3}\) |
The Trigonometry Connection
The exact sine and cosine values for \(30^\circ\), \(45^\circ\), and \(60^\circ\) come directly from special right triangles.
To see the connection, scale each special triangle so its hypotenuse equals \(1\). The horizontal leg becomes cosine, and the vertical leg becomes sine.
Visual Connection
Exact Values
These values should be understood from the triangle ratios rather than treated as unrelated facts.
| Angle | Triangle | \(\sin\theta\) | \(\cos\theta\) | \(\tan\theta\) |
|---|---|---|---|---|
| \(30^\circ\) | \(30\text{-}60 \text{-}90\) | \(\dfrac{1}{2}\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{\sqrt{3}}{3}\) |
| \(45^\circ\) | \(45\text{-}45 \text{-}90\) | \(\dfrac{\sqrt{2}}{2}\) | \(\dfrac{\sqrt{2}}{2}\) | \(1\) |
| \(60^\circ\) | \(30\text{-}60 \text{-}90\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{1}{2}\) | \(\sqrt{3}\) |
Worked Example 6
Use a special right triangle to find the exact value of \(\sin 60^\circ\).
The opposite side is the long leg, so its length is \(\sqrt{3}\).
Check Your Reasoning
Most mistakes happen because a side is matched with the wrong part of the ratio. Label the angles and side types before calculating.
The ratio \(1:1:\sqrt{2}\) belongs to a \(45\text{-}45\text{-}90\) triangle.
In a \(30\text{-}60\text{-}90\) triangle, \(x\) is opposite \(30^\circ\), not \(60^\circ\).
If the hypotenuse is known, you usually divide to work backward toward the basic value \(x\).
An answer such as \(\sqrt{72}\) should be simplified before it is treated as a final answer.
Ready to Practice?
Practice identifying special triangles, matching sides with angles, finding missing lengths, simplifying radicals, and using the ratios to find exact trigonometric values.
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