Right Triangles & Trigonometry
Solving Right Triangles
Finding missing sides and angles of a right triangle using trigonometric ratios.
Prerequisites
- Sine, Cosine & Tangent
Finding Every Missing Piece
A right triangle has a 90° angle, one known side, and one known acute angle. Before reading on, think about this: with just those two known values, do you think it's possible to find every other side and angle in the entire triangle?
It is — a right triangle's third angle is automatically 90° minus the known acute angle (since all three angles sum to 180°, and one is already 90°), and every trig ratio connects the known side to each of the others, one at a time. The last piece is knowing how to go the other direction: finding an angle when two sides are already known, using the inverse trig functions (sin⁻¹, cos⁻¹, tan⁻¹) that undo sine, cosine, and tangent, the same 'undo an operation' logic used throughout this course for square roots and other inverse relationships.
Worked Example — Finding All Missing Sides Given an Angle and a Side
Worked Example — Finding a Missing Angle Using an Inverse Trig Function
Worked Example — An Application: Angle of Elevation
Tip
Common Mistakes
Using a calculator in the wrong angle mode (radians instead of degrees, or vice versa), producing a nonsensical result.
Check the calculator's mode setting before starting any trigonometric calculation — a right triangle problem given in degrees needs the calculator set to degree mode throughout.
Choosing the wrong trig ratio for the two known sides, such as using sine when the known sides are actually the two legs (which calls for tangent).
Identify exactly which two sides are known or wanted relative to the angle — opposite and hypotenuse calls for sine, adjacent and hypotenuse calls for cosine, opposite and adjacent calls for tangent.
Key Takeaways
- A right triangle is fully solvable from any one side and any one acute angle, or from any two sides.
- Inverse trig functions (sin⁻¹, cos⁻¹, tan⁻¹) find a missing angle from a known ratio of sides.
- Real applications, like angle of elevation problems, translate directly into a right triangle with a known angle and side.
Summary
Solving right triangles completes the toolkit connecting angles and side lengths. The next unit turns to circles, where angles relate to arcs in equally structured ways.
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