Grade by Grade
C. Trigonometric Functions · Trigonometry of Acute Triangles

Choosing the Sine Law or the Cosine Law

C1.4About 70 minutes
Learning goals
  • I can describe the conditions that tell me whether to use the primary ratios, the sine law or the cosine law.
  • I can classify given information as AAS, ASA, SSA, SAS or SSS.
  • I can choose the most efficient path to solve an acute triangle.
  • I can explain my choice in words.

Words to know

AAS / ASA
Two angles and a side. Always gives a complete opposite pair once you use the angle sum: sine law.
SSA
Two sides and an angle opposite one of them: sine law (finds an angle).
SAS
Two sides and the angle between them: cosine law (finds the third side).
SSS
Three sides: cosine law (finds an angle).

Picking the right tool

You now have three tools for triangles: the primary ratios (right triangles only), the sine law and the cosine law. Expectation C1.4 asks you to describe the conditions that tell you which one to use. A good choice makes a problem short; a bad one can leave you stuck.

The key question is: do I know a side together with the angle across from it?

Is there a right angle?YesSOH CAH TOA,PythagorasNo (acute triangle)Is there a completeopposite pair?Yes: SINE LAWAAS, ASA (after anglesum), SSANo: COSINE LAWSAS (find a side)SSS (find an angle)Then finishwith whicheverlaw is easiest.
A decision chart for solving any triangle in this course.
Known partsNameComplete opposite pair?Start with
two angles and a side not between themAASyessine law
two angles and the side between themASAafter the angle sumsine law
two sides and an angle opposite one of themSSAyessine law (for an angle)
two sides and the angle between themSASnocosine law (for a side)
three sidesSSSnocosine law (for an angle)
three angles onlyAAAno side knowncannot find sides: many similar triangles fit
Summary of the conditions for each law.
SAScosine lawSSScosine lawASAsine lawSSAsine lawThick sides and double arcs are the KNOWN parts.
The four cases you will meet most. The cosine-law cases (left two) have no side known together with its opposite angle.
Worked example

Example 1: Two angles and a side

In △ABC, ∠A = 58°, ∠B = 57° and AB = 32 m. Find BC.

  1. Known: two angles and the side between them (ASA). Choose the sine law.
  2. ∠C = 180° − 58° − 57° = 65°. AB = c = 32 is across from ∠C: complete pair.
  3. BC = a = 32 sin 58° ÷ sin 65° ≈ 29.9 m.

BC ≈ 29.9 m

Worked example

Example 2: Two sides and the angle between them

In △PQR, PQ = 5.6 cm, PR = 7.3 cm and ∠P = 49°. Find QR.

  1. ∠P is between PQ and PR: SAS. There is no complete pair. Choose the cosine law.
  2. QR² = 5.6² + 7.3² − 2(5.6)(7.3) cos 49° ≈ 31.36 + 53.29 − 53.64 ≈ 31.01
  3. QR ≈ 5.57 cm

QR ≈ 5.6 cm

Worked example

Example 3: Two sides and an angle not between them

In acute △ABC, b = 15 cm, c = 13 cm and ∠B = 71°. Find ∠C.

  1. ∠B is across from b, so (b, ∠B) is a complete pair: SSA. Choose the sine law.
  2. sin C = 13 sin 71° ÷ 15 ≈ 0.8194
  3. ∠C ≈ 55.0°

∠C ≈ 55°

Worked example

Example 4: Three sides

A triangular lot has sides 23 m, 19 m and 25 m. Find the angle between the 19 m and 25 m sides.

  1. SSS: choose the cosine law. The angle between 19 and 25 is across from 23.
  2. cos θ = (19² + 25² − 23²) ÷ (2 × 19 × 25) = (361 + 625 − 529) ÷ 950 = 457 ÷ 950 ≈ 0.4811
  3. θ ≈ 61.2°

θ ≈ 61°

Efficiency tips

  • If the triangle has a right angle, SOH CAH TOA is fastest.
  • Use the angle sum whenever two angles are known; it is free.
  • After the cosine law gives you a side or an angle, you usually have a complete pair: switch to the sine law (it is quicker).
  • Use given values over calculated ones when you have a choice, to avoid rounding error.
Just for fun

Choosing between the sine law and the cosine law is like choosing a key on a keyring: the wrong one doesn't break anything, it just won't turn. If there is no complete pair, the sine law simply won't turn.

Hands-on activity

Triangle sorting game

You need: 12 index cards (or scraps of paper), a pen, and a partner if possible.

  1. On each card, draw a triangle and mark three known parts (sides with a tick, angles with an arc). Make some AAS, ASA, SSA, SAS and SSS, and one with a right angle.
  2. Shuffle the cards. Sort them into three piles: SOH CAH TOA, sine law and cosine law.
  3. For each card, say the reason out loud ("complete pair: b and ∠B", or "angle between the two sides: SAS").
  4. Pick two cards from the cosine-law pile, give them numbers, and solve them completely.

What to look for: every SAS and SSS card in the cosine-law pile, every AAS, ASA and SSA card in the sine-law pile, and the right-triangle card with SOH CAH TOA (although the laws also work there).

Your turn

1

Name the case and the law to start with: a, b and ∠C known.

2

Name the case and the law to start with: ∠A, ∠C and b known.

3

Name the case and the law to start with: a, c and ∠A known.

4

In △DEF, d = 8.5 m, e = 9.9 m and f = 7.2 m. Find ∠E.

5

In △KLM, ∠K = 63°, ∠L = 52° and k = 18 cm. Find l.

6

In △STU, s = 30 m, t = 26 m and ∠U = 62°. Find u.

7

Why can't you find the side lengths of a triangle if you only know its three angles?

More practice: warm-up

1

Which law needs a complete opposite pair?

2

SSS: which law?

3

AAS: which law?

4

A right triangle: which tool is fastest?

5

Two angles are 48° and 77°. Find the third.

More practice: on your own

1

Solve acute △ABC: a = 11, b = 13, ∠C = 64°.

2

Solve △PQR: ∠P = 47°, ∠R = 68°, q = 40.

3

Sara wants to start △ABC (b = 7, c = 9, ∠A = 55°) with the sine law: b/sin B = c/sin C. Explain why she is stuck.

4

Solve the triangle with sides 14, 15 and 16.

5

Write a one-sentence rule a classmate could use to choose between the sine law and the cosine law.

Challenge

1

Solve △ABC with ∠A = 50°, b = 10 and c = 12 in two ways: (1) cosine law for a, then sine law for ∠B; (2) cosine law for a, then cosine law for ∠B. Compare.

2

Can the cosine law be used for an AAS problem? Explain.

3

Show that the sine law and the cosine law agree on the triangle with sides 6, 7 and 8 by finding ∠A (across from 6) both ways after you find ∠C ≈ 75.5° (across from 8).

Check your understanding

Quiz

1.

You know two sides and the angle between them. Start with:

2.

You know ∠A, ∠B and side c. What should you do first?

3.

Which set of givens CANNOT determine a unique triangle size?

4.

Which case uses the sine law to find an ANGLE?

5.

In △PQR, PQ = 5.6, PR = 7.3 and ∠P = 49°. QR ≈

6.

After using the cosine law to find a side in SAS, the quickest way to find another angle is usually:

For parents and teachers

Big idea: Expectation C1.4 says students should describe the conditions that guide when to use the sine law or the cosine law. Being able to say why matters as much as the calculation.

How to help: Before your teen calculates, ask: "Which case is this (AAS, ASA, SSA, SAS or SSS)? Is there a complete pair?" The sorting game in the activity is a quick way to practise.

Common difficulties:

  • Starting with the sine law in SAS and getting an equation with two unknowns.
  • Not noticing that ASA becomes a sine-law problem once the third angle is found.
  • Using SOH CAH TOA in a triangle that has no right angle.