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Trigonometry · Trigonometry: Similar Triangles, Right Triangles and Acute Triangles

Solving Right Triangles

2.2About 70 minutes
Learning goals
  • I can use standard labelling (a opposite ∠A) for the sides and angles of a triangle.
  • I can find all unknown sides and angles of a right triangle.
  • I can choose between the Pythagorean theorem, the primary trigonometric ratios and the angle sum.
  • I can avoid rounding errors by using unrounded values.

Words to know

solve a triangle
Find all the unknown sides and angles.
standard labelling
Vertices and their angles use capital letters (A, B, C); the side opposite each angle uses the same small letter (a, b, c).
Pythagorean theorem
In a right triangle with hypotenuse c, a² + b² = c².
complementary angles
Two angles that add to 90°. The two acute angles of a right triangle are complementary.

Putting the tools together

A right triangle has three sides and three angles. One angle is already known (90°). If you know two more pieces, and at least one of them is a side, you can find everything else. That is called solving the triangle. You now have three tools:

  1. the angle sum: the two acute angles add to 90°;
  2. the Pythagorean theorem: a² + b² = c² (from Grades 8 and 9);
  3. the primary trigonometric ratios and their inverses.

Standard labelling

Mathematicians label a triangle so that each side is named after the angle opposite it: side a is opposite ∠A, side b is opposite ∠B and side c is opposite ∠C. In this unit, if ∠C = 90°, then c is the hypotenuse. You will use this labelling for the sine law and cosine law later in the unit.

AaBbCcCapital letter = angle (vertex)Small letter = side opposite ita is opposite ∠Ab is opposite ∠Bc is opposite ∠C (the hypotenuse here)
Standard labelling: small letters for sides, capital letters for the opposite angles.
What do you know?Two sides• third side: Pythagorean theorem• one angle: sin⁻¹, cos⁻¹ or tan⁻¹• other angle: 90° − first angleOne side and one acute angle• other angle: 90° − given angle• each other side: sin, cos or tan• check: Pythagorean theorem
Which tool first? It depends on what you are given. Always finish with a quick check.
Worked example

Example 1: Given two legs

Solve △ABC with ∠C = 90°, a = 6 cm and b = 8 cm. Round angles to the nearest degree.

  1. Hypotenuse: c² = 6² + 8² = 36 + 64 = 100, so c = 10 cm.
  2. ∠A: a is opposite and b is adjacent. tan A = 6/8, so ∠A = tan⁻¹(0.75) ≈ 36.9° ≈ 37°.
  3. ∠B = 90° − 36.87° ≈ 53°.
  4. Check: the bigger angle (53°) is opposite the longer leg (8). ✓

c = 10 cm, ∠A ≈ 37°, ∠B ≈ 53°

Worked example

Example 2: Given an angle and the hypotenuse

Solve △ABC with ∠C = 90°, ∠A = 28° and c = 15 m. Round sides to one decimal place.

  1. ∠B = 90° − 28° = 62°.
  2. a is opposite ∠A: a = 15 sin 28° = 7.042... ≈ 7.0 m.
  3. b is adjacent to ∠A: b = 15 cos 28° = 13.244... ≈ 13.2 m.
  4. Check with Pythagoras (unrounded values): 7.042² + 13.244² ≈ 49.6 + 175.4 = 225 = 15². ✓

∠B = 62°, a ≈ 7.0 m, b ≈ 13.2 m

AaBbC1528°?Aa ≈ 7.0Bb ≈ 13.2C1528°62°GivenSolved
Example 2 before and after solving, drawn to scale. Once one acute angle and one side are known, every other part follows.
Worked example

Example 3: Given an angle and a leg

Solve △ABC with ∠C = 90°, ∠B = 55° and b = 9 cm.

  1. ∠A = 90° − 55° = 35°.
  2. From ∠B, b = 9 is opposite and a is adjacent: tan 55° = 9/a, so a = 9 ÷ tan 55° = 6.301... ≈ 6.3 cm.
  3. c is the hypotenuse: sin 55° = 9/c, so c = 9 ÷ sin 55° = 10.986... ≈ 11.0 cm.
  4. Check: c is the longest side, and the larger angle (55°) is opposite the longer leg (9 > 6.3). ✓

∠A = 35°, a ≈ 6.3 cm, c ≈ 11.0 cm

Worked example

Example 4: Given the hypotenuse and a leg

A right triangle has hypotenuse 17 cm and one leg 8 cm. Find the other leg and both acute angles.

  1. Other leg: √(17² − 8²) = √(289 − 64) = √225 = 15 cm.
  2. The angle opposite the 8 cm leg: sin⁻¹(8/17) ≈ 28.1° ≈ 28°.
  3. Other angle: 90° − 28.07° ≈ 62°.

15 cm; angles ≈ 28° and 62°

Avoid rounding errors

When one answer is used to find another, use the unrounded value (store it in calculator memory, or go back to the original data). Better still, calculate each unknown from the given information whenever you can. In Example 2, finding b with √(15² − 7.0²) using the rounded 7.0 gives 13.3, not the correct 13.2.

Just for fun

Why was the right triangle so calm during the test?

Because it always knew one of its angles was right.

Hands-on activity

Solve a triangle from your home

You need: a measuring tape, a calculator, and a right-angled corner with a diagonal you can measure (a rectangular table top, a door, a window or a book cover).

  1. Measure the length and width of the rectangle. They are the two legs of a right triangle; the diagonal is the hypotenuse.
  2. Predict the diagonal with the Pythagorean theorem and the angle between the diagonal and the long side with tan⁻¹.
  3. Now measure the diagonal corner to corner. Then check the angle with a protractor.
  4. Calculate your percent error for the diagonal: |measured − predicted| ÷ predicted × 100%.

What to look for: a percent error under about 2% for the diagonal and an angle within 1° or 2°. For a 76 cm by 152 cm table the diagonal is about 170 cm and the angle is tan⁻¹(76 ÷ 152) ≈ 27°.

Your turn

1

Solve △ABC with ∠C = 90°, a = 5 and b = 12.

2

Solve △ABC with ∠C = 90°, ∠A = 40° and c = 20.

3

Solve △ABC with ∠C = 90°, ∠B = 63° and a = 7.

4

Solve △ABC with ∠C = 90°, c = 25 and a = 7.

5

Solve △ABC with ∠C = 90°, ∠A = 52° and a = 10.

6

Check your answer to the second question with the Pythagorean theorem.

More practice: warm-up

1

What do the two acute angles of a right triangle add to?

2

In right △ABC, ∠C = 90° and ∠A = 35°. Find ∠B.

3

The legs are 9 cm and 12 cm. Find the hypotenuse.

4

The hypotenuse is 26 and one leg is 10. Find the other leg.

5

What does it mean to "solve a triangle"?

More practice: on your own

1

Solve △PQR with ∠Q = 90°, PQ = 14 cm and QR = 9 cm.

2

Solve △XYZ with ∠Z = 90°, ∠X = 18° and XY = 40 m.

3

A right triangle has a 71° angle, and the leg adjacent to it is 3.5 cm. Find the other two sides and the other acute angle.

4

In Example 2, Noor found a = 7.0, then b = √(15² − 7.0²). What did she get, and why is it different from 13.2?

5

An isosceles right triangle has a hypotenuse of 10 cm. How long is each leg?

Challenge

1

A right triangle has an area of 24 cm², and one leg is 6 cm. Solve the triangle.

2

A right triangle has a 30° angle and its shortest side is 4 cm. Find its perimeter, to one decimal place.

3

A box is 4 cm long, 3 cm wide and 12 cm tall. Find the angle between the long diagonal of the box (from a bottom corner to the opposite top corner) and the bottom of the box.

Check your understanding

Quiz

1.

A right triangle has legs 9 cm and 12 cm. What is its hypotenuse?

2.

In right △ABC, ∠C = 90° and ∠A = 34°. What is ∠B?

3.

In right △ABC with ∠C = 90°, you know a and b. Which expression gives ∠A directly?

4.

∠C = 90°, ∠A = 28° and c = 15. What is b?

5.

Why should you use unrounded values in later steps?

6.

A right triangle has hypotenuse 17 and one leg 8. What is the other leg?

For parents and teachers

Big idea: Expectation 2.2: determining all the sides and angles of right triangles using the primary trigonometric ratios and the Pythagorean theorem.

How to help: Have your teen sketch and label the triangle (a opposite A and so on) before calculating, and tick off each unknown as they find it. A check at the end (Pythagoras, or "biggest angle opposite longest side") catches most mistakes.

Common difficulties:

  • Using rounded answers in later steps.
  • Mixing up which leg is opposite and which is adjacent after switching angles.
  • Forgetting the simplest step: the two acute angles add to 90°.
  • Subtracting in the wrong order in the Pythagorean theorem (leg² = hypotenuse² − other leg²).