What is the Sine Rule?
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
1. What is the meaning and use of cosine, sine and tangent in mathematics?
\[\textrm{Cosine,}\] \[\textrm{Sine}\] and \[\textrm{tangent}\] is a part of trigonometry. It is known as the function of an angle. The use of these functions is to calculate unknown angles and distances from available or given angles.
\[\textrm{Sine}\] : \[\textrm{Sine (Sin)}\] function is defined as the ratio of the side opposite the angle \[\textrm{(Perpendicular)}\] to the \[\textrm{hypotenuse.}\]
\[\textrm{Sin}\] = \[\frac{Perpendicular}{Hypotenuse}\]
\[\textrm{Cosine}\] : \[\textrm{Cosine (Cos)}\] function is defined as the ratio of \[\textrm{Adjacent (Base)}\] to the hypotenuse.
\[\textrm{cos}\] = \[\frac{Base}{Hypotenuse}\]
\[\textrm{Tangent}\] : \[\textrm{Tangent}\] function, defined as the ratio of the opposite \[\textrm{(perpendicular)}\] side to the \[\textrm{adjacent (base)}\] of a right angled triangle.
\[\textrm{tan}\] = \[\frac{Perpendicular}{Base}\]
\[\textrm{Cosecant,}\] \[\textrm{Secant}\] and \[\textrm{Cotangent}\] are the reciprocals of \[\textrm{Sine,}\]
\[\textrm{Cosine}\]
and \[\textrm{tangent}\]
respectively.
2. Can trigonometry be used in everyday life? If yes, what are its practical uses?
Yes, trigonometry is used in everyday life but not directly. Here are some examples of it:
Sound engineers use trigonometry for the development of computer music. How? So as you may be aware, sound travels in waves, and this pattern, while not as regular as a sine or cosine function, is essential in the development of computer music. Because a computer cannot listen to or perceive music in the same way that we can, it is mathematically represented by its constituent sound waves. As a result, sound engineers must understand the fundamentals of trigonometry. And the excellent music created by these sound professionals is utilized to soothe us from our hectic, stressful lives - all due to trigonometry.
Apart from this, trigonometry functions are used to measure the height of a mountain or building and are also used by flight engineers. Moreover, crime scene investigators also use trigonometry to investigate the scene and make assumptions based on it.
3. Can we calculate the area of a triangle with the sine rule?
Yes, it is possible to calculate the area of a triangle with the sine cosine rule.
You need to remember the basic formula of the triangle area that is \[\textrm{A}\] = \[\frac{1}{2}\] \[\textrm{bh}\]
, where \[\textrm{b}\] and \[\textrm{h}\] are the base and height, respectively.
Consider the following example for better understanding:
Let’s take a triangle \[\textrm{ABC}\] with the following measurement:
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\[\textrm{AB = c = 3 cm}\]
\[\textrm{BC = a = 4 cm}\]
\[\textrm{Angle B}\] = \[145^{\circ}\]
Therefore, by using the formula for area of triangle \[\textrm{A}\] = \[\frac{1}{2}\] \[\textrm{ac sin B}\] , where a and c are the lengths of the sides.
Formula: \[\frac{a}{sin A}\] = \[\frac{b}{sin B}\] = \[\frac{c}{sin C}\]
Putting the values in the formula,the area,
\[\textrm{A}\] = \[\frac{1}{2}\] \[\times\] \[\textrm{4}\] \[\times\] \[\textrm{3}\] \[\times\] \[\textrm{sin}\] \[145^{\circ}\]
\[\textrm{R = 6 sin}\] \[145^{\circ}\]
= \[\textrm{6 (0.5736)}\]
= \[\textrm{3.44}\]
Hence, the area of triangle \[\textrm{ABC}\] is \[\textrm{3.44 sq.cm.}\]
