9.1 Sine Rule
In everyday life, many situations involve non-right-angled triangles where the Pythagoras Theorem and basic right-angled trigonometric ratios cannot be applied directly. The sine rule provides a powerful method to find unknown sides and angles for any type of triangle when specific sets of information are given.
Learning Objectives
- Make and verify conjectures on the relationship between the ratios of side lengths of a triangle and the sine of the opposite angles.
- State and use the sine rule formula: \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\).
- Solve non-right-angled triangles involving the sine rule given two angles and one side, or two sides and a non-included angle.
- Determine the conditions for the existence of an ambiguous case and calculate the two possible sets of solutions.
- Solve practical real-world problems involving triangles using the sine rule, including angles of elevation and bearings.
Summary of Conditions for Triangle Existence and the Ambiguous Case
Compare how the length of the opposite side \(a\) relative to height \(h\) and adjacent side \(c\) determines whether 0, 1, or 2 triangles exist.
| Condition | Geometric Meaning | Number of Triangles |
|---|---|---|
| \(a < h\) | Side \(a\) is too short to reach the base line. | No triangle exists |
| \(a = h\) | Side \(a\) touches the base line at exactly one point, forming a right angle. | Exactly one right-angled triangle |
| \(h < a < c\) | Side \(a\) can swing to cut the base line at two different points. | Two distinct triangles (ambiguous case) |
| \(a \ge c\) | Side \(a\) intersects the base line only once to the right of \(A\). | Exactly one triangle |
How to Check for the Ambiguous Case
Identify the given acute angle \(A\), adjacent side \(c\), and opposite side \(a\).
Calculate the perpendicular height \(h = c \sin A\).
Compare \(a\) with \(h\) and \(c\).
If \(h < a < c\), find the second possible angle using \(\angle C_1 = 180^\circ - \angle C\).
Short Explanation
The sine rule expresses that in any triangle, the ratio between the length of any side and the sine of its opposite angle is constant.
1. Derivation of the Sine Rule
In \(\triangle ABC\), draw a perpendicular line \(CD\) to the base \(AB\) such that \(CD = h\). In \(\triangle BCD\), \(\sin B = \dfrac{h}{a} \implies h = a \sin B\). In \(\triangle ACD\), \(\sin A = \dfrac{h}{b} \implies h = b \sin A\). Equating both expressions for \(h\) gives \(a \sin B = b \sin A\), so \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B}\).
2. The Sine Rule Formula
For any triangle \(ABC\): \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\) or \(\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}\).
3. Conditions for Applying the Sine Rule
To solve a triangle using the sine rule, either of the following conditions must be known: two angles and the length of one side (AAS or ASA), or the lengths of two sides and a non-included angle (SSA).
4. Solving Unknown Sides and Angles
When finding an unknown side length, write the side in the numerator: \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B}\). When finding an unknown angle, write the sine of the angle in the numerator: \(\dfrac{\sin A}{a} = \dfrac{\sin B}{b}\). For any triangle, \(\angle A + \angle B + \angle C = 180^\circ\).
5. The Ambiguous Case
An ambiguous case occurs when two sides and a non-included acute angle are given, leading to two possible distinct triangles from the same given measurements. Given acute angle \(\angle A\), adjacent side \(c\), and opposite side \(a\), with perpendicular height \(h = c \sin A\): if \(a < h\), no triangle exists; if \(a = h\), exactly one right-angled triangle exists; if \(a \ge c\), exactly one triangle exists; if \(h < a < c\), two distinct triangles exist.
6. Characteristics of the Ambiguous Case
Two different triangles can be formed: \(\triangle ABC\) and \(\triangle AB_1C\). If the first calculated angle is acute (\(\angle C\)), the second possible angle is obtuse: \(\angle C_1 = 180^\circ - \angle C\). Corresponding remaining angles: \(\angle B = 180^\circ - \angle A - \angle C\) and \(\angle B_1 = 180^\circ - \angle A - \angle C_1\).
7. Application to Real-World Situations
Bearings are measured clockwise from North (\(000^\circ\) to \(360^\circ\)). Angles of elevation and heights are applied in surveying, finding distances to towers, kites, flag posts, or across rivers.
8. Key Insight
When given two sides and a non-included acute angle, be alert to the ambiguous case. Check the perpendicular height \(h = c \sin A\). If \(h < a < c\), side \(a\) can swing to two different positions on the base, forming one acute-angled triangle and one obtuse-angled triangle.
The Sine Rule
\(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\)or equivalently: \(\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}\)
Try to Answer First
Answer in your mind, then press "Check Answer".
In \(\triangle PQR\), \(\angle P = 48^\circ\), \(\angle Q = 80^\circ\), and side \(PR = 14.3\text{ cm}\). Find the length of side \(QR\) (denoted as \(x\)) correct to three decimal places.
In \(\triangle XYZ\), \(\angle X = 55.5^\circ\), \(YZ = 6.2\text{ cm}\), and \(XZ = 7.05\text{ cm}\). Find the acute angle \(\angle Y\) (denoted as \(x\)) correct to two decimal places.
In \(\triangle ABC\), \(\angle BAC = 40^\circ\), \(AB = 20\text{ cm}\), and \(BC = 14\text{ cm}\). (a) Determine whether the ambiguous case exists. (b) Find the two possible values of \(\angle C\).
Common Mistakes
- Incorrect pairing of sides and angles: pairing a side with an adjacent angle instead of the angle strictly opposite to it.
- Placing unknown values in the denominator: writing \(\dfrac{\sin A}{a} = \dfrac{\sin B}{b}\) when solving for side \(a\), which leads to algebraic inversion errors. Always place the unknown variable in the numerator.
- Overlooking the ambiguous case: finding only the acute angle and failing to check whether an obtuse angle \(180^\circ - \theta\) is also possible when two sides and a non-included angle are given.
- Incorrect bearing format: forgetting that three-figure bearings must always be measured clockwise from North, e.g. writing \(39.13^\circ\) instead of \(039.13^\circ\).
Concept Misunderstandings
The sine rule only applies to non-right-angled triangles.
The sine rule actually works for all triangles, including right-angled triangles. Since \(\sin 90^\circ = 1\), the sine rule directly simplifies to the definition of \(\sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}\).
An ambiguous case exists whenever two sides and an angle are given.
The ambiguous case only exists when the given angle is a non-included acute angle and the opposite side \(a\) satisfies \(h < a < c\). If \(a \ge c\), only one triangle exists. If \(a < h\), no triangle exists.
Both possible angles in the ambiguous case can always form a triangle.
The second angle (\(180^\circ - \theta\)) must leave a positive remaining angle when added to the original given angle, i.e. \(\angle A + \angle C_1 < 180^\circ\).
Summary
- The Sine Rule states: \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\) or \(\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}\).
- Used when given: two angles and one side, or two sides and a non-included angle.
- For an acute angle \(A\) with adjacent side \(c\) and height \(h = c \sin A\): \(a < h \implies\) no triangle; \(a = h \implies\) one right-angled triangle; \(a \ge c \implies\) one triangle; \(h < a < c \implies\) two triangles (ambiguous case), where \(\angle C_1 = 180^\circ - \angle C\).
Short Activity
Objective: Test your understanding of the sine rule, the ambiguous case, and key terms.
A. Objective Quiz
1 Under which of the following conditions can the sine rule be used directly to solve a triangle?
2 In \(\triangle ABC\), \(\angle A = 30^\circ\), side \(c = 10\text{ cm}\), and side \(a = 4\text{ cm}\). How many triangles can be formed with these dimensions?
B. Fill in the Blanks
3 For any triangle \(ABC\), the sine rule states that \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\_\_\_\_\_}\).
4 An ambiguous case occurs when the given non-included angle is acute and the opposite side \(a\) satisfies the condition \(h < a < \_\_\_\_\_\), where \(h\) is the height of the triangle.
C. Matching / Drag and Drop
Match each geometric condition regarding side \(a\), height \(h\), and adjacent side \(c\) with the correct outcome of triangle formation. On a phone, tap a term first, then tap the matching box.
