Solving right triangles (mixed contexts)
Solving Right Triangles
Solving Right Triangles is a topic in Right Triangles & Trigonometry in the Common Core State Standards. It is aligned to Standard G-SRT.8, which requires students to use trigonometric ratios and the Pythagorean theorem to solve right triangles in applied problems.
Solving a right triangle finds its unknown sides and angles using the trigonometric ratios, their inverses, and the Pythagorean theorem.
Theory
To solve a right triangle is to find every missing side and angle. The tools:
- a trig ratio when you know one side and one acute angle;
- the Pythagorean theorem and an inverse trig function when you know two sides;
- the acute angles are complementary (sum \(90^\circ\)).
Real problems use the angle of elevation (looking up) or angle of depression (looking down) from the horizontal.
The solving tools:
How to solve a right triangle
- Sketch and label the given parts.
- Side from angle: pick the trig ratio and solve.
- Angle from sides: use an inverse trig function.
- Finish with the Pythagorean theorem and complementary angles.
Use sine: opposite over hypotenuse.
| \(\text{opp}\) | \(=\) | \(20\sin 50^\circ\) |
| \(\approx\) | \(15.3\) |
Use inverse tangent.
| \(\theta\) | \(=\) | \(\tan^{-1}\!\dfrac{9}{12}\) |
| \(\approx\) | \(36.9^\circ\) |
Height is opposite; distance is adjacent, so use tangent.
| \(h\) | \(=\) | \(30\tan 40^\circ\) |
| \(\approx\) | \(25.2\ \text{ft}\) |
The two acute angles are complementary.
| \(90^\circ-35^\circ\) | \(=\) | \(55^\circ\) |
Common pitfalls
Frequently asked questions
What does it mean to solve a right triangle?
To find all of its unknown sides and angles from the information given.
What is the angle of elevation?
The angle measured upward from the horizontal to a line of sight; the angle of depression is measured downward.
Which trig ratio finds a height from a distance and an angle?
Tangent, since height is opposite and distance is adjacent: height \(=\) distance \(\times\tan\theta\).
How do you find the second acute angle?
Subtract the known acute angle from \(90^\circ\); the acute angles are complementary.