5.3 Trigonometry
Key Takeaways
- Label opposite, adjacent, and hypotenuse from the chosen angle.
- Use SOH-CAH-TOA to select sine, cosine, or tangent.
- Use Pythagoras when two right-triangle sides are known.
- Check that side lengths and angles are physically plausible.
Trigonometry in Right-Angled Triangles
Trigonometry is the study of the relationships between the sides and angles of triangles. In aircraft maintenance, it is an indispensable tool used to calculate structural dimensions, resolve aerodynamic forces, and perform navigation vector calculations.
1. Trigonometric Ratios
For any right-angled triangle, we define the three primary ratios relative to an angle $\theta$:
- Sine ($\sin\theta$): $\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$
- Cosine ($\cos\theta$): $\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
- Tangent ($\tan\theta$): $\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}$
These ratios are commonly remembered using the acronym SOH-CAH-TOA.
2. Pythagoras' Theorem
For a right-angled triangle with legs of length $a$ and $b$, and a hypotenuse of length $c$, the relationship is defined by:
Worked Example (Symmetry and Squareness Check): During heavy maintenance, a technician must perform a symmetry check to verify that the wings and fuselage are properly aligned and have not been deformed by hard landings. The technician measures the diagonal distances from the centerline of the fuselage at the tail to the left and right wingtips. If the structure is symmetrical, these diagonal distances must be identical.
Additionally, to verify if a fuselage frame jig is perfectly square ($90^\circ$), the technician uses the 3-4-5 rule (a practical application of Pythagoras' theorem: $3^2 + 4^2 = 9 + 16 = 25 = 5^2$). By marking points at 3 feet and 4 feet along the adjacent frame members, the diagonal distance between them must measure exactly 5 feet if the corner is square.
Worked Example (Strut Length Calculation): A landing gear door support strut is installed at an angle of $30^\circ$ relative to a vertical structural bulkhead. If the vertical distance from the bulkhead attachment pivot to the level of the door hinge is 24 inches (adjacent side), what is the required length of the strut (hypotenuse)?
- Identify the trigonometric relationship: $\cos(30^\circ) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{24}{L}$.
- Solve for the strut length $L$:
3. Standard Trigonometric Values
Technicians should be familiar with the values of trigonometric functions for standard angles:
| Angle ($\theta$) | $\sin\theta$ | $\cos\theta$ | $\tan\theta$ |
|---|---|---|---|
| $0^\circ$ | $0$ | $1$ | $0$ |
| $30^\circ$ | $0.5$ | $\frac{\sqrt{3}}{2} \approx 0.8660$ | $\frac{1}{\sqrt{3}} \approx 0.5774$ |
| $45^\circ$ | $\frac{1}{\sqrt{2}} \approx 0.7071$ | $\frac{1}{\sqrt{2}} \approx 0.7071$ | $1$ |
| $60^\circ$ | $\frac{\sqrt{3}}{2} \approx 0.8660$ | $0.5$ | $\sqrt{3} \approx 1.732$ |
| $90^\circ$ | $1$ | $0$ | Undefined |
A structural support strut is installed at an angle of 30 degrees relative to a vertical bulkhead. If the vertical distance (adjacent side) between the attachment points is 24 inches, what is the required length of the strut (hypotenuse)?
In a right triangle, which ratio is opposite divided by hypotenuse?
A right triangle has legs of 3 and 4 units. What is its hypotenuse?
You've completed this section
Continue exploring other exams