Sine Rule which is also known as the Law of Sine, gives the relationships between sides and angles of any triangle. Sine Rule is a powerful tool in trigonometry that can be used to find solutions for triangles using various properties of triangles. With the help of the Sine rule, we can find any side of a triangle with ease only if we are given the angles and any of the sides of the triangle. In this article, we will explore all the aspects of the sine rule, including its formula, proof, and applications. Other than these all topics, we will also learn how to solve problems based on the sine rule or law of sine. So, let's learn about the sine rule in this article.
Sine Rule Definition
The Sine Rule or the Law of Sines is a mathematical relationship that relates the ratios of the sides of a triangle to the sines of its opposite angles (non-right-angled triangle). In other words, it states that the ratio of the side to the sin of the opposite angle to the side always remains constant. Sine Rule is used to solving triangles when the measures of some angles and their corresponding opposite sides are known. Let's learn about the mathematical expression or formula for the sine rule.
Formula for Sine Rule
Let a, b, and c be the lengths of the three sides of a triangle ABC and A, B, and C by their respective opposite angles. Now the expression for the Sine Rule is given as,
sin A/a = sin B/b = sin C/c = k
a/sin A = b/sin B = c/sin C = k
Sine Rule Proof
In triangle ABC, the sides of the triangle are given by AB = c, BC = a and AC = b.
Let us draw a perpendicular BD that is perpendicular to AC. Now we have two right-angles triangles ADC and BDC.
BD = h is the height of the triangle ABC.
In triangle ADC,
sin A = h/c ⇢ (1)
In triangle BDC,
sin C = h/a ⇢ (2)
Now by dividing the equations (1) and (2).
We get, sin A/sin C = a/c ⇒ a/sin A = c/sin C ⇢ (3)
Similarly, draw a perpendicular AE that is perpendicular to BC. Now AEB and BEC are the right angled triangles separated by h2.
In triangle AEB,
sin B = H/c ⇢ (4)
In triangle AEC,
sin C = Hsolve/b ⇢ (5)
Now by dividing equations (4) and (5),
sin B/sin C = b/c ⇒ b/sin B = c/sin C ⇢ (6)
Now by equating the equations (3) and (6) we get,
a/sin A = b/sin B = c/sin C
sin A/a = sin B/b = sin C/c
Solving Triangles using the Law of Sine
With the help of the law of sine, we can calculate the unknown angle or sides of the triangle in some cases which we will discuss in the following headings:
Finding Unknown Side Lengths
To find the unknown side length of a triangle when one side or any two angles of the triangle is given, we can use the following steps:
Step 1: Check whether you have enough information to use the Law of Sines. You need one side and two angles including the opposite angle to the given side.
Step 2: Write down the formula for the Law of Sines: sin(A)/a = sin(B)/b = sin(C)/c, where A, B, and C are angles and a, b, and c are the lengths of the opposite sides, respectively.
Step 3: Substitute the known values into the equation.
Step 4: Solve the equation for the unknown side by taking those two parts of the equation containing the known and unknown sides.
Let's consider an example to understand the process better.
Example: In a triangle, ABC has ∠A measuring 30°, ∠B measuring 120°, and side b measuring 10 units, then find the length of side c (opposite side to ∠C).
Solution:
Step 1: We have ∠A and ∠B, as well as side b. We need to find side c, which is the side opposite ∠C(=30°).
Thus, we have enough information to calculated the length of side c.
Step 2: The formula for the Law of Sines is sin(A)/a = sin(B)/b = sin(C)/c.
Step 3: Substitute the known values into the equation.
sin(B)/b = sin(C)/c
⇒ sin(120°)/10 = sin(30°)/c
We have sin(A)/a = sin(B)/b. Plugging in the values, we get sin(30)/c = sin(60)/10.
Step 4: Solve the equation for the unknown side.
sin(120°)/10 = sin(30°)/c
⇒ sin(30°)/10 = sin(30°)/c
⇒ c = 10
Therefore, in this example, the length of side c is 10 units.
Finding Unknown Angles
To find the unknown angle of any triangle, we can use similar steps as the above. But there are some minor changes as we discuss:
Step 1: Check whether you have enough information to use the Law of Sines. You need one angle and two sides including the opposite side to the given angle.
Step 2: Write down the formula for the Law of Sines: sin(A)/a = sin(B)/b = sin(C)/c, where A, B, and C are angles and a, b, and c are the lengths of the opposite sides, respectively.
Step 3: Substitute the known values into the equation.
Step 4: Solve the equation for the unknown angle by taking those two parts of the equation containing the known and unknown angles.
Let's consider an example of a solved problem to understand the use of the sine rule to find the unknown angles:
Example: In triangle ABC, ∠C is 40°, side c is 8 units long, and side b is 12 units long. Find ∠A.
Solution:
Step 1: Check whether you have enough information to use the Law of Sines. We have ∠C and two sides, side b and side c, including the side opposite to angle A. So we have enough information to proceed.
Step 2: Write down the formula for the Law of Sines: sin(A)/a = sin(B)/b = sin(C)/c.
Step 3: Substitute the known values into the equation. We have ∠C= 40°, b = 8 units, and c = 12 units.
So we can write sin(B)/8 = sin(40°)/12.
Step 4: Solve the equation for the unknown angle by taking those two parts of the equation containing the known and unknown angles.
sin(B)/12 = sin(40°)/8
⇒ sin(B) = 12 × sin(40°)/8
⇒ sin(B) ≈ 0.93 [sin 40° ≈ 0.64]
Now, let's find angle C using the inverse sine function:
B ≈ sin-1(0.93)
⇒ B ≈ 68.45°
Since we know that the sum of angles in a triangle is 180 degrees, we can find angle B:
A = 180 - (AB+ C)
⇒ A = 180 - (40 + 68.45)
⇒ A ≈ 71.55°
Therefore, in triangle ABC, angle A is approximately 71.55°
Limitation of Sine Rule
There are various limitations of the Sine Rule, which are listed as follows:
- Applicability: As sine rule is defined for non-right angle triangles, it can only be used for the same. Thus sine rule is a very specific rule which only applies for non-right angle triangles and not to other polygons.
- Triangle Requirement: To use the sine rule, we need one side and its opposite angle in the triangle and if those conditions are not met, we can't able to use sine rule.
- Ambiguity: The Sine Rule can lead to ambiguous results in certain cases. Specifically, when solving for an angle, the Sine Rule may yield two possible solutions, one acute and one obtuse. Additional information or constraints are needed to determine the correct solution.
- Dependence on angles: The Sine Rule is primarily based on the relationship between the angles of a triangle. It is not as useful for determining the lengths of the sides directly. In some cases, the measurements of the angles may be more difficult to obtain or less precise than the lengths of the sides.
- Precision: The Sine Rule may introduce rounding errors and imprecisions, especially when dealing with triangles with very small or very large angles or sides. Truncating or rounding values during calculations can result in inaccuracies.
- Ambiguity in obtuse triangles: The Sine Rule is not well-defined for obtuse triangles (triangles with one angle greater than 90 degrees). In such cases, the rule can still be used, but it requires modification and consideration of additional factors.
Sample Problems on Sine Rule
Problem 1: Find the remaining lengths of the triangle XYZ when ∠X = 30° and ∠Y = 45° and x = 5 cm.
Solution:
Given data, ∠X = 30°, ∠Y = 45° and x = 5 cm
We know that sum of the three angles of a triangle is 180°
So, ∠X + ∠Y + ∠Z = 1
30° + 45° + ∠Z = 180° ⇒ 75° + ∠Z = 180°
∠Z = 105°
Now from the law of Sines, x/ sin X = y/ sin Y = z/ sin Z
x/sin X = y/ sin Y ⇒ 5/ sin 30° = y/ sin 45°
⇒ x/(1/2) = y/(1/√2) ⇒ 10 =√2y ⇒ y = 7.07 cm
Similarly, x/ sin X = z/sin Z ⇒ 5/sin 30° = z/sin 105° [sin 105° = (√6 + √2)/4 = 0.965]
⇒ 5/(1/2) = z/(0.965) ⇒ z = 9.65 cm
Problem 2: Find ∠P and ∠Q and the length of the third side when ∠R = 36° and p = 2.5 cm and r = 7 cm.
Solution:
Given, ∠R = 36° and p = 2.5 cm and r = 7 cm
From the Sine Rule Formula we have,
p/sin P = q/sin Q = r/sin R
⇒ 2.5/sin P = q/ sin Q = 7/sin 36°
⇒ 2.5/sin P = 7/sin 36° [sin 36° = 0.5878]
⇒ sin P = 0.20992 ⇒ P = sin-1 (0.20992)
⇒ ∠P = 12.12°
⇒ We have, ∠P + ∠Q + ∠R = 180°
⇒12.12° + ∠Q + 36° = 180° ⇒ ∠Q = 131.88°
⇒ q/sin 131.88° = 7/sin 36°
⇒ q/0.7445 = 7/0.5878 ⇒ q = 8.866 cm (approximately)
Hence ∠P = 12.12°, ∠Q = 131.88° and q = 8.866 cm
Problem 3: Find the ratio of the sides of the triangle ABC when ∠A = 15°, ∠B = 45°, and ∠C = 120°.
Solution:
Given: ∠A =15°, ∠B = 45° and ∠C = 120°
From the Sine Rule formula, a/ sin A = b/ sin B = c/ sin C ⇒ a : b : c = sin A : sin B : sin C
⇒ a : b : c = sin 15°: sin 45° : sin 120°
sin 15° = (√3 - 1)/2√2 = 0.2588 (approximate value)
sin 45° = 1/√2 = 0.7071 (approximate value)
sin 120° = √3/2 = 0.866 (approximate value)
Hence the ratio of the three sides of the triangle ABC is a : b : c = 0.2588 : 0.7071 : 0.866
Problem 4: Find the area of the triangle ABC when BC = 10 cm, AB = 12 cm, and ∠B= 30°.
Solution:
Given, BC = a = 10 cm, AB = c = 12 cm and ∠B= 30°
We know that, Area of the triangle = ½ (base) (height) = ½ (a) (h) ⇢ (1)
From the figure, sin B = height/c
h = c sin B ⇢ (2)
Now substitute equation (2) in (1),
Area of the triangle ABC = ½ (a)(c) sin B= ½ (10) (12) sin 30° [sin 30° = ½]
⇒ Area = ½ (120) ½ = 30 cm2
Hence the area of the triangle ABC is 30 cm2.
Problem 5: Find ∠ACB if a = 3 cm, c = 1 cm, and ∠BAC = 60°.
Solution:
Given, ∠BAC = 60°, a = 3 cm and c = 1 cm
From the Sine Rule Formula, we have sin A/a = sin C/c
⇒ sin 60°/3 = sin C/1 [sin 60° = √3/2]
⇒ (√3/2)/3 = sin C/3 ⇒ sin C = 1/(2√3) ⇒ sin C = 0.2887 (approximately)
⇒ ∠C = sin-1(0.2887) ⇒ ∠C = 16.77°
Hence ∠ACB = 16.77°
Problem 6: Find the length of the side YZ, if the area of the triangle XYZ is 24 cm2, ∠Y = 45°, and z = 4 cm.
Solution:
Given, Area of the triangle XYZ = 24 cm2, ∠y = 45° and z = 4 cm
From the Sine rule law, we have Area of triangle XYZ = ½ (x)(z) sin Y
⇒ 24 = ½ (x)(4) sin 45° [sin 45° = 1/√2]
⇒ 24 = (x) × (2) × (1/√2)
⇒ 12√2 = x ⇒ x = 16.968 cm
Hence the length of the side YZ = x = 16.968 cm
Problem 7: Find q when ∠P= 108°, ∠Q = 43°and p = 15 cm.
Solution:
Given, ∠P = 108° , ∠Q = 43° and p = 15 cm
From the Sine Rule Formula we have, p/sin P = q/sin Q
⇒ 15/sin 108° = q/sin 43°
⇒ 15/(0.9510) = q/(0.6820) [sin 108° = 0.9510 & sin 43° = 0.6820]
⇒ q = 10.757 cm (approximately)
Hence q =10.57 cm
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