You can find an angle inside a triangle using different methods depending on the information you already have, such as other angles or the lengths of sides.
Understanding the properties of triangles is key to finding unknown angles. The most fundamental property is that the sum of the interior angles in any triangle is always 180 degrees.
Based on the information provided, here are two primary ways to find an angle:
Method 1: Using Known Angles
If you know the measure of two angles within a triangle, finding the third angle is straightforward.
- Rule: If two angles are known, the third angle is found by adding the two known angles and subtracting that sum from 180.
Let the three angles of a triangle be Angle A, Angle B, and Angle C.
If you know Angle A and Angle B, you can find Angle C using the formula:
Angle C = 180° - (Angle A + Angle B)
Example:
Suppose a triangle has two angles measuring 70° and 50°. What is the measure of the third angle?
- Add the two known angles: 70° + 50° = 120°
- Subtract the sum from 180°: 180° - 120° = 60°
Therefore, the third angle measures 60°.
Method 2: Using Known Sides (Trigonometry)
When you know the lengths of two sides of a right triangle and need to find an angle, you can use trigonometric ratios.
- Rule: If two sides are known, the formula SOH CAH TOA can be applied.
SOH CAH TOA is a mnemonic to remember the definitions of the three basic trigonometric ratios: Sine, Cosine, and Tangent. These relate the angles of a right triangle to the ratios of its sides.
- SOH: Sine = Opposite / Hypotenuse
- CAH: Cosine = Adjacent / Hypotenuse
- TOA: Tangent = Opposite / Adjacent
| Acronym | Ratio | Formula |
|---|---|---|
| SOH | Sine (sin) | Opposite side / Hypotenuse |
| CAH | Cosine (cos) | Adjacent side / Hypotenuse |
| TOA | Tangent (tan) | Opposite side / Adjacent side |
Here, O stands for the length of the side Opposite the angle you are trying to find, A is the length of the side Adjacent to the angle (but not the hypotenuse), and H is the length of the Hypotenuse (the side opposite the right angle).
To find the angle itself using these ratios, you need to use the inverse trigonometric functions:
- Inverse Sine:
Angle = sin⁻¹(Opposite / Hypotenuse) - Inverse Cosine:
Angle = cos⁻¹(Adjacent / Hypotenuse) - Inverse Tangent:
Angle = tan⁻¹(Opposite / Adjacent)
Example:
Consider a right triangle where you know the length of the side opposite an angle is 8 units and the length of the adjacent side is 6 units. You want to find the angle.
- You know the Opposite and Adjacent sides. The TOA ratio uses these sides:
tan(Angle) = Opposite / Adjacent. - Plug in the values:
tan(Angle) = 8 / 6 = 4/3. - Use the inverse tangent function to find the angle:
Angle = tan⁻¹(4/3). - Using a calculator,
Angle ≈ 53.13°.
Therefore, the angle is approximately 53.13 degrees.
These two methods, utilizing the sum of angles property or trigonometric ratios based on side lengths, are fundamental ways to determine unknown angles within a triangle.
