How to Apply the Sine Rule in Solving Triangles
It is easy to find the missing angle when the other two angles of a triangle are given, but the real confusion begins when you have to find a missing angle when given one angle and two sides or vice-versa. Don’t worry; this doesn’t mean it's impossible. Our mathematicians have got it covered! In the 11th century, Ibn Mu'adh al-Jayyani presented the sine rule, also known as the law of sines.
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But before jumping on the law of sines, let's understand the meaning of the term sine.
Let’s take a right triangle $\text{XYZ}$
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Considering that $\text{XZ}$ is the right triangle's hypotenuse, the sine of the angle $\text{YZA}$ equals the$\frac{\text{Length XY}}{\text{Length XZ}}$ ratio.
Sine < $\text{YZA}$=$\frac{\text{XY}}{\text{ XZ}}$
Likewise, the sine of angle $\text{YXZ}$ equals the $\frac{\text{Length YZ}}{\text{Length XZ}}$ ratio.
Therefore, $\text{YXZ}$= $\frac{\text{YZ}}{\text{XZ}}$
As a result, the sine of an angle is the ratio of the angle's opposing side length to the length of the hypotenuse.
What is Law of Sines?
Sine rule is one of the most useful mathematical laws, especially when it comes to solving triangles. The sine rule calculator helps to calculate the sides and angles of a triangle when not enough information is provided.
The rule of sines connects the side length ratios of triangles to their opposing angles. This proportion is true for all three sides and opposing angles. We can use the sine rule in triangle to get the missing angle or side of any triangle using the existing data.
Definition of The Law of Sines
The ratio of a triangle's side and matching angle equals the diameter of the triangle's circumcircle. As a result, the sine law may be written as, $\frac{\text{a}}{\sin A} =\frac{\text{b}}{\sin B}=\frac{\text{c}}{\sin C} $
where a,b and c are the lengths of the sides of a triangle and A, B and C are the opposite angles.
To use the area of triangle sine rule, one must know either two angles and one side of the triangle or two sides and one angle. Along with this, two sides and an angle opposite one of them can be used to calculate the angle through the sine rule.
How to Use the Law of sines?
If either one angle and the matching opposite side of a triangle are known, and either the side or angle of another side is also given, the Law of Sines may locate the missing information. The greatest approach to learn how is to practice using an example. Consider the triangle below. Using what we've learnt, we'll solve for side "c."
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Here the given items are:
B= 35°
C= 105°
b = 7
The first step is to identify what is given and note it down in a crisp form
The second step is to determine what needs to be calculated; we have to find side “c”.
The next step is to put all the given numbers into the sine equation, which is going to be:
$\frac{\text{a}}{\sin A} =\frac{7}{\sin (35^{\circ})}=\frac{\text{c}}{\sin(105)}$
By applying cross multiplication and replacing the values for sin B and sin C with numerical estimates, and simply find the angle and corresponding side by solving it.
Explanation of The Sine Rule Proof
First, identify the triangle ABC in the normal manner so that angle A is the opposite side a.
Angle B is the opposite of side b, while angle C is the opposite of side c.
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Using basic trigonometry,
$\sin(B) = \frac{\text{h}}{\text{c}}$ & $\sin(c) = \frac{\text{h}}{\text{b}}$
Therefore, h= c sin (B) & h= b sin (c) respectively.
So, c sin (B) = b sin (C)
Dividing both sides by bc, we get: $\frac{\sin (B)}{b} =\frac{\sin (C)}{c}$
Similarly, using an altitude from B or C, we would also include $\frac{\sin (A)}{a}$
Thus, the sine rule: $\frac{\sin (A)}{a}= \frac{\sin (B)}{b} =\frac{\sin (C)}{c}$
Which can be written as: $\frac{a}{\sin (A)}= \frac{b}{\sin {B}} =\frac{c}{\sin {C}}$
Hence the law of sine proved!
FAQs on Sine Rule Explained: Definition, Formula & Uses
1. What is the Sine Rule and what is its primary purpose in trigonometry?
The Sine Rule, also known as the Law of Sines, is a fundamental principle in trigonometry that establishes a relationship between the sides of a triangle and the sines of their opposite angles. Its primary purpose is to find unknown side lengths or angles in non-right-angled triangles (oblique triangles), where basic ratios like SOH CAH TOA do not directly apply.
2. What is the formula for the Sine Rule?
The Sine Rule can be expressed in two equivalent forms. For a triangle with sides a, b, c and opposite angles A, B, C respectively, the formula is:
- To find a side length: a/sin(A) = b/sin(B) = c/sin(C)
- To find an angle: sin(A)/a = sin(B)/b = sin(C)/c
3. Under what specific conditions should the Sine Rule be used to solve a triangle?
The Sine Rule is the correct tool to use when you know:
- Two angles and any one side (AAS or ASA): If you know two angles, you can find the third (since angles sum to 180°), and then use the rule to find the unknown sides.
- Two sides and a non-included angle (SSA): This is when you know the lengths of two sides and the angle opposite one of them. It's important to be cautious in this case as it can lead to the 'ambiguous case'.
4. What is the key difference in application between the Sine Rule and the Cosine Rule?
The key difference lies in the information you are given about the triangle.
- Use the Sine Rule when you have a 'pair' of information – an angle and its opposite side (e.g., angle A and side a), plus one other piece of information (another side or angle). This applies to AAS, ASA, and SSA cases.
- Use the Cosine Rule when you do not have a corresponding side-angle pair. This is for cases where you know two sides and the angle between them (SAS) or all three sides (SSS).
5. How is the Sine Rule derived or proven?
The Sine Rule can be proven by dropping a perpendicular (height 'h') from one vertex to the opposite side. For a triangle ABC, if we drop a perpendicular from vertex C to side AB, we create two right-angled triangles. In one triangle, sin(A) = h/b, so h = b sin(A). In the other, sin(B) = h/a, so h = a sin(B). Since 'h' is the same for both, we can equate the expressions: b sin(A) = a sin(B). Rearranging this gives a/sin(A) = b/sin(B). The same logic can be applied using another vertex to complete the full rule.
6. What is the 'ambiguous case' of the Sine Rule and why does it occur?
The ambiguous case occurs when using the Sine Rule with two known sides and a non-included angle (SSA). It arises because the sine of an angle and the sine of its supplement (180° - angle) are equal. For example, sin(30°) = 0.5 and sin(150°) = 0.5. When you solve for an unknown angle using the arcsin function, your calculator will only give you the acute angle. However, an obtuse angle might also be a valid solution, leading to the possibility of zero, one, or two different triangles that fit the given SSA criteria.
7. Can you provide a real-world example of how the Sine Rule is used?
A classic real-world application of the Sine Rule is in surveying and navigation. Imagine needing to find the distance to a ship at sea from two points on the shore, A and B. You can measure the distance between A and B. Then, from point A, you measure the angle to the ship, and from point B, you do the same. You now have a triangle where you know one side (the distance between A and B) and two angles. This is an ASA (Angle-Side-Angle) case, perfect for using the Sine Rule to calculate the distances from A and B to the ship without ever leaving the shore.
