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

Choosing the Right Tool: Solving Acute Triangles

3.3 2.2About 75 minutes
Learning goals
  • I can decide whether to use the primary trigonometric ratios, the sine law or the cosine law.
  • I can solve an acute triangle completely, finding all unknown sides and angles.
  • I can combine the cosine law, the sine law and the angle sum in one solution.
  • I can check a solved triangle for reasonableness.

Words to know

AAS / ASA
Two angles and a side are known (the side is either not between them, or between them).
SSA
Two sides and an angle opposite one of them are known.
SAS
Two sides and the angle between them (the contained angle) are known.
SSS
All three sides are known.

Three tools, one decision

You now own three tools for triangles: SOH CAH TOA (with the Pythagorean theorem) for right triangles, the sine law and the cosine law. The hardest part of most problems is not the arithmetic; it is choosing the right tool. This lesson gives you a reliable way to decide, then puts all three tools together to solve acute triangles completely.

Is there a right angle?yesSOH CAH TOA+ Pythagorean theoremnoIs there a complete pair?(side + opposite angle)yesSine lawAAS, ASA (after 180°), SSAnoSAS or SSSCosine lawthen sine law or cosine lawTwo angles known butno pair? Find the thirdangle first (180°).
A decision chart. Ask the questions in order: right angle? complete pair? Otherwise it must be SAS or SSS.
AASSSASASSSSsine lawcosine law
The thick sides and angles are the given information. With a complete opposite pair (AAS, SSA), start with the sine law. With SAS or SSS there is no pair, so start with the cosine law.
Worked example

Example 1: SAS, solved completely

Solve △ABC with a = 11 cm, b = 14 cm and ∠C = 62°.

  1. SAS (∠C is between a and b). Start with the cosine law: c² = 11² + 14² − 2(11)(14) cos 62° = 317 − 308 cos 62° = 172.40...
  2. c = √172.40... = 13.130... ≈ 13.1 cm. Store the unrounded value.
  3. Now c and ∠C are a complete pair. Use the sine law for the angle opposite the shorter of the other two sides (a = 11): sin A = 11 sin 62° ÷ 13.130... = 0.7397..., so ∠A ≈ 47.7°.
  4. Angle sum: ∠B = 180° − 62° − 47.7° = 70.3°.
  5. Check: the longest side (b = 14) is opposite the largest angle (∠B ≈ 70°). ✓

c ≈ 13.1 cm, ∠A ≈ 48°, ∠B ≈ 70°

Aa = 11Bb = 14Cc ≈ 13.162°48°70°1. Cosine law → c ≈ 13.12. Sine law → ∠A ≈ 48°3. Angle sum → ∠B ≈ 70°Check: longest side bis opposite the largest angle B.
Example 1 drawn to scale, with every part found. Three tools in three steps.
Worked example

Example 2: SSS, solved completely

Solve the triangle with sides a = 5 m, b = 6 m and c = 7 m.

  1. SSS: cosine law. Start with the largest angle, ∠C, opposite c = 7: cos C = (25 + 36 − 49) ÷ 60 = 0.2, so ∠C ≈ 78.5°. It is less than 90°, so the triangle is acute.
  2. ∠A (opposite 5): cos A = (36 + 49 − 25) ÷ 84 = 60 ÷ 84 ≈ 0.7143, so ∠A ≈ 44.4°.
  3. ∠B = 180° − 78.5° − 44.4° ≈ 57.1°.
  4. Check: 44.4° < 57.1° < 78.5° matches 5 < 6 < 7. ✓

∠A ≈ 44°, ∠B ≈ 57°, ∠C ≈ 78°

Worked example

Example 3: AAS, solved completely

Solve △ABC with ∠A = 42°, ∠B = 71° and a = 8 cm.

  1. Complete pair: a and ∠A. Sine law.
  2. ∠C = 180° − 42° − 71° = 67°.
  3. b = 8 sin 71° ÷ sin 42° = 11.30... ≈ 11.3 cm.
  4. c = 8 sin 67° ÷ sin 42° = 11.00... ≈ 11.0 cm.
  5. Check: the smallest angle (42°) is opposite the shortest side (8 cm). ✓

∠C = 67°, b ≈ 11.3 cm, c ≈ 11.0 cm

Solving tips

  • After a cosine-law step, the sine law is usually quicker for the next angle. In acute triangles it always gives the right angle. It is good practice to use it for the angle opposite the shorter side; you will see why in Grade 11.
  • Find the last angle with the angle sum (180°), not with another law.
  • Keep unrounded values in your calculator. Round only the answers you write down.
  • Final check: the order of the sides (shortest to longest) must match the order of their opposite angles.
Just for fun

Choosing between the sine law and the cosine law is like choosing a key for a lock. The trick is not to try every key on the ring: just look at the lock (what you are given) first.

Hands-on activity

Triangle-solving card game

You need: 12 index cards (or pieces of paper), a pencil and a partner.

  1. On each card, sketch a triangle and mark three given parts, with numbers. Make 3 of each type: right triangle, AAS, SAS and SSS. Keep every angle under 90° (except the right angles). For SSS, choose sides where the square of the longest side is less than the sum of the squares of the other two, so the triangle is acute.
  2. Shuffle the cards. Your partner draws one and has 10 seconds to say which tool to start with and why.
  3. Then both of you solve the triangle completely and compare answers.
  4. Score 1 point for the right first tool and 2 points for a fully correct solution.

What to look for: quick, correct tool choices ("there is no complete pair, so cosine law"), all six parts found, and a final check that the longest side is opposite the largest angle.

Your turn

1

You know ∠A, ∠B and a. Which tool do you start with?

2

You know a, b and ∠C. Which tool?

3

You know a, b and c. Which tool?

4

∠C = 90°, and you know a and ∠A. Which tool?

5

Solve △ABC with ∠A = 50°, b = 9 cm and c = 11 cm.

6

Solve the triangle with sides 8, 9 and 10.

7

Solve △ABC with ∠B = 38°, ∠C = 77° and b = 14 m.

More practice: warm-up

1

Which tool works only in right triangles?

2

What is a "complete pair"?

3

What do the angles of a triangle add to?

4

You know two sides and an angle opposite one of them. Which law?

5

After you know two angles of a triangle, how do you find the third?

More practice: on your own

1

Solve △PQR with p = 12, q = 15 and r = 13.

2

Solve △ABC with ∠A = 75°, ∠C = 45° and b = 20 cm.

3

Solve △XYZ with x = 16, y = 18 and ∠Z = 53°.

4

Why can you switch to the sine law after using the cosine law in an SAS problem?

5

Eli tried to start an SAS problem with the sine law and got stuck. Explain why.

Challenge

1

Solve acute △ABC with b = 9 cm, a = 7 cm and ∠B = 64°.

2

An isosceles triangle has two sides of 12 cm and a base of 10 cm. Find all three angles.

3

A triangle has sides 13, 14 and 15. Use the cosine law to find the angle opposite the side of length 13, then find the area of the triangle exactly.

Check your understanding

Quiz

1.

You know two sides and the contained angle. Which tool do you start with?

2.

You know ∠A, ∠B and side a. Which tool?

3.

A right triangle has a known hypotenuse and a known acute angle. Which is the most direct tool?

4.

After using the cosine law to find the third side in an SAS problem, what is a good next step?

5.

In △ABC, ∠A = 42° and ∠B = 71°. What is ∠C?

6.

A triangle has sides 5, 6 and 7. About how large is its largest angle?

For parents and teachers

Big idea: Expectation 3.3: determining the measures of sides and angles in acute triangles using the sine law and the cosine law (and choosing between them and the right-triangle tools).

How to help: Play the card game; saying the reason out loud ("no complete pair, so cosine law") builds the decision habit. For each solved triangle, ask "Is the longest side opposite the largest angle?"

Common difficulties:

  • Starting with the sine law in SAS or SSS situations.
  • Using a rounded side from the cosine law in the sine law step.
  • Forgetting to find the third angle first in ASA problems.
  • Leaving a triangle half-solved: "solve" means all three sides and all three angles.