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AI HL · 12.10

Geometry & Trigonometry

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12.10.13

The sine rule: the ambiguous case

12.10.13

The sine rule: the ambiguous case

Target: I can use the sine rule to find an angle when I know two sides and an angle opposite one of them. (+3 more)

By the end of this section

Key points

The sine rule for an angle

  • In triangle ABC: \(\frac{\sin A}{a}=\frac{\sin B}{b}\). Side a is opposite angle A.
  • Use it when you know an angle, the side opposite it and one more side. Put the unknown angle on top: \(\sin B=\frac{b\sin A}{a}\).
  • Always sketch. Mark which side is opposite which angle.

Two angles from one sine

  • Two different angles between 0° and 180° have the same sine: \(\sin\theta=\sin(180^\circ-\theta)\). The calculator's \(\sin^{-1}\) gives only the acute one.
  • Second solution: \(180^\circ-\) the calculator angle. Keep it only if (given angle) + (second angle) < 180°.
  • If the given angle is obtuse, the second solution never fits. The same is true when the side opposite the given angle is longer than the other side.

How many triangles? (side a opposite the given angle A, other side b)

  • Work out \(h=b\sin A\), the shortest distance from C to the base line.
  • \(a<h\): no triangle (\(\sin B\) would be more than 1).
  • \(a=h\): one triangle, right-angled at B.
  • \(h<a<b\): two triangles. This is the ambiguous case.
  • \(a\geq b\): one triangle.

On the calculator

Inverse sine and the second angle

  • Check the calculator is in degree mode.
  • Work out the sine first: \(\sin B=\frac{b\sin A}{a}\). Then use \(\sin^{-1}\) of that number.
  • Subtract the answer from 180° for the second angle.
  • A value above 1 gives an error: no triangle.

Screenshot and video to come

Explore: the ambiguous case

You know angle A, the side b next to it and the side a opposite it. Side a swings from C like a compass. Change A, a and b, and watch where the dashed arc meets the base line.



Number of triangles

Change a side or the angle.

  1. Make a smaller and smaller. At what value does the arc stop reaching the base line? What is that value in terms of b and A?
  2. Press "Set a = b sin A". What kind of triangle is it? What is angle B?
  3. Keep a between b sin A and b. Why are there two triangles? What do the two values of B add up to?
  4. Make a longer than b. Why does one of the two triangles disappear? Try making A bigger than 90°.

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
One angle only

In triangle ABC, angle A = 68°, BC = 12.4 cm and AC = 9.6 cm. Find the angle ABC, θ.

12.4 cm9.6 cm68°θABC
Working
  1. Sine rule

    \(\frac{\sin\theta}{9.6}=\frac{\sin68^\circ}{12.4}\)

    BC (12.4) is opposite the 68°. AC (9.6) is opposite θ.

  2. Rearrange

    \(\sin\theta=\frac{9.6\sin68^\circ}{12.4}=0.7178\)

    From the calculator, in degree mode.

  3. Calculator angle

    \(\theta=45.87^\circ\)

    This is the acute angle.

  4. Second angle?

    \(180^\circ-45.87^\circ=134.1^\circ,\;\;134.1+68=202.1>180\)

    Too big. Reject it.

  5. Answer

    \(\theta=45.9^\circ\)

    One triangle.

Video walkthrough coming
Two triangles

In triangle ABC, angle A = 38°, BC = 11 cm and AC = 15.3 cm. Find the two possible values of angle ABC, θ. The diagram shows both triangles.

11 cm15.3 cm11 cm38°AB′BC
Working
  1. Sine rule

    \(\frac{\sin\theta}{15.3}=\frac{\sin38^\circ}{11}\)

    BC (11) is opposite 38°. AC (15.3) is opposite θ.

  2. First angle

    \(\sin\theta=0.8563,\;\;\theta=58.91^\circ\)

    From the calculator.

  3. Second angle

    \(\theta=180^\circ-58.91^\circ=121.1^\circ\)

    Same sine, obtuse angle.

  4. Check both

    \(58.91+38=96.91<180,\;\;121.1+38=159.1<180\)

    Both leave room for a third angle. Both triangles exist.

  5. Answer

    \(\theta=58.9^\circ\text{ or }121^\circ\)

    The 11 cm side can swing to two places on the base line, B and B′.

Video walkthrough coming
Counting triangles

In triangle ABC, angle A = 30° and AC = 14 cm. How many different triangles are there when BC is (a) 5 cm, (b) 7 cm, (c) 10 cm, (d) 15 cm?

Working
  1. The height

    \(h=14\sin30^\circ=7\)

    The shortest distance from C to the base line through A.

  2. (a) a = 5

    5 < 7: no triangle

    BC is too short to reach the base line. In the sine rule, sin B would be more than 1.

  3. (b) a = 7

    7 = 7: one triangle

    BC just touches the base. Angle B = 90°.

  4. (c) a = 10

    7 < 10 < 14: two triangles

    Long enough to reach the base, shorter than AC, so it can swing two ways.

  5. (d) a = 15

    15 > 14: one triangle

    BC is longer than AC. The second position falls on the wrong side of A.

Video walkthrough coming