✅ Sine Formula ⭐️⭐️⭐️⭐️⭐️

To calculate the angle of the right-angled triangle, sine formula is used. The relation between the sides and angles of the right angle is shown through this formula. Sine is the ratio of the opposite side to the hypotenuse side of the right triangle. The longest side is the hypotenuse and the opposite side of the hypotenuse is the opposite side. In trigonometry, Sin is the shorthand of sine function.

The Sine Angle Formula is,

Solved Examples

Question 1: Calculate the sine angle of a right triangle whose opposite side and hypotenuse are 10 cm and 12 cm respectively?

Solution:

Given, Opposite side = 10 cm Hypotenuse = 12 cm

Using the formula:

Question 2: If sin A = 0.5, then find the value of x from the following figure.

Solution:
Given,
Sin A = 0.5 = 5/10 = ½
We know that sin θ = Opposite/Hypotenuse
BC/AC = ½
12/AC = ½
AC = 12 × 2 = 24 cm
By Pythagoras theorem,

AC² = AB² + BC²
24² = x² + 12²
x² = 576 – 144
x² = 432
x = √452 cm

x=20.78 cm

Sine, Cosine and Tangent

Sine, Cosine and Tangent (often shortened to sin, cos and tan) are each a ratio of sides of a right angled triangle:

For a given angle θ each ratio stays the same
no matter how big or small the triangle is

To calculate them:

Divide the length of one side by another side

Size Does Not Matter

The triangle can be large or small and the ratio of sides stays the same.

Only the angle changes the ratio.

Try dragging point “A” to change the angle and point “B” to change the size:

Good calculators have sin, cos and tan on them, to make it easy for you. Just put in the angle and press the button.

But you still need to remember what they mean!

In picture form:

Sohcahtoa

Examples

Why?

Why are these functions important?

• Because they let us work out angles when we know sides
• And they let us work out sides when we know angles

Less Common Functions

To complete the picture, there are 3 other functions where we divide one side by another, but they are not so commonly used.

They are equal to 1 divided by cos, 1 divided by sin, and 1 divided by tan:

The Law of Sines

The Law of Sines (or Sine Rule) is very useful for solving triangles:

And it says that:

When we divide side a by the sine of angle A
it is equal to side b divided by the sine of angle B,
and also equal to side c divided by the sine of angle C

Sure … ?

Well, let’s do the calculations for a triangle I prepared earlier:

The answers are almost the same!
(They would be exactly the same if we used perfect accuracy).

So now you can see that:

How Do We Use It?

Let us see an example:

Finding an Unknown Angle

In the previous example we found an unknown side …

… but we can also use the Law of Sines to find an unknown angle.

In this case it is best to turn the fractions upside down (sin A/a instead of a/sin A, etc):

Sometimes There Are Two Answers !

There is one very tricky thing we have to look out for:

Two possible answers.

Imagine we know angle A, and sides a and b. We can swing side a to left or right and come up with two possible results (a small triangle and a much wider triangle) Both answers are right!

This only happens in the “Two Sides and an Angle not between” case, and even then not always, but we have to watch out for it.

Just think “could I swing that side the other way to also make a correct answer?”

But wait! There’s another angle that also has a sine equal to 0.9215…

The calculator won’t tell you this but sin(112.9°) is also equal to 0.9215…

So, how do we discover the value 112.9°?

Easy … take 67.1° away from 180°, like this:

180° − 67.1° = 112.9°

So, always check to see whether the alternative answer makes sense.

• … sometimes it will (like above) and there are two solutions
• … sometimes it won’t (see below) and there is one solution

We looked at this triangle before.

As you can see, you can try swinging the “5.5” line around, but no other solution makes sense.

So this has only one solution.

What Is the Sine formula?

The sine function of a right triangle is the ratio of its perpendicular to its height. Thus, sine formula can be expressed as,

sin ⁡θ = P/H

Where,

• P = perpendicular
• H = hypotenuse

Let us see the applications of sine formula in the below solved examples.

Solved Examples Using Sine formula

What is Law of Sines?

The law of sines relates the ratios of side lengths of triangles to their respective opposite angles. This ratio remains equal for all three sides and opposite angles. We can therefore apply the sine rule to find the missing angle or side of any triangle using the requisite known data.

Law of Sines: Definition

The ratio of the side and the corresponding angle of a triangle is equal to the diameter of the circumcircle of the triangle. The sine law is can therefore be given as,

a/sinA = b/sinB = c/sinC = 2R

• Here a, b, c are the lengths of the sides of the triangle.
• A, B, and C are the angle of the triangle.
• R is the radius of the circumcircle of the triangle.

Law of Sines Formula

The law of sines formula is used for relating the lengths of the sides of a triangle to the sines of consecutive angles. It is the ratio of the length of the side of the triangle to the sine of the angle thus formed between the other two remaining sides. The law of sines formula is used for any triangle apart from SAS triangle and SSS triangle. It says,

a/sin A = b/sin B = c/sin C

where,

• a, b, and c are the lengths of the triangle
• A, B, and C are the angles of the triangle.

This formula can be represented in three different forms given as,

• a/sinA = b/sinB = c/sinC
• sinA/a = sinB/b = sinC/c
• a/b = sinA/sinB; a/c = sinA/sinC; b/c = sinB/sinC

Example: Given a = 20 units c = 25 units and Angle C = 42º. Find the angle A of the triangle.

Solution:

For the given data, we can use the following formula of sine law: a/sinA = b/sinB = c/sinC
⇒ 20/sin A = 25/sin 42º
⇒ sin A/20 = sin 42º/25
⇒ sin A = (sin 42º/25) × 20
⇒ sin A = (sin 42º/25) × 20
⇒ sin A = (0.6691/5) × 4
⇒ sin A = 0.5353
⇒ A = sin^-1(0.5363)
⇒ A = 32.36º

Answer: ∠A = 32.36º

Proof of Law of Sines Formula

The law of sines is used to compute the remaining sides of a triangle, given two angles and a side. This technique is known as triangulation. It can also be applied when we are given two sides and one of the non-enclosed angles. But, in some such cases, the triangle cannot be uniquely determined by this given data, called the ambiguous case, and we obtain two possible values for the enclosed angle. To prove the sine law, we consider two oblique triangles as shown below.

In the first triangle, we have:

h/b = sinA
⇒ h = b sinA

In the second triangle, we have:

h/a = sinB
⇒ h = a sinB

Also, sin(180º – B) = sinB

Equalizing the h values from the above expressions, we have:
a sinB = b sinA
⇒ a/sinA = b/sinB

Similarly, we can derive a relation for sin A and sin C.
asinC = csinA
⇒ a/sinA = c/sinC

Combining the above two expressions, we have the following sine law.
a/sinA = b/sinB = c/sinC

Tips and Tricks on Law of Sines

• The triangulation technique is used to find the sides of a triangle when two angles and one side of a triangle is known. For this the sine law is helpful.
• This sine law of trigonometry should not be confused with the sine law in physics.
• Further deriving from this sine law we can also find the area of an oblique triangle.
  Area of a triangle = (1/2) ab sinC = (1/2) bc sinA = (1/2) ca sinB
• Also sine law provides a relationship with the radius R of the circumcircle,a/sinA = b/sinB = c/sinC = 2R
• Cosine law: This proves a relationship between the sides and one angle of a triangle,c² = a² + b² – 2ab⋅cos C
• Tangent law: This has been derived from the sine law and it gives the relationship between the sides and angles of a triangle.

Applications of Sine Law

The law of sines finds application in finding the missing side or angle of a triangle, given the other requisite data. The sine law can be applied to calculate:

• The length of the side of a triangle using ASA or AAS criteria.
• The unknown angle of a triangle.
• The area of the triangle.

Ambiguous Case of Law of Sines

While applying the law of sines to solve a triangle, there might be a case when there are two possible solutions, which occurs when two different triangles could be created using the given information. Let us understand this ambiguous case while solving a triangle using Sine law using the following example.

Example: If the side lengths of △ABC are a = 18 and b = 20 with ∠A opposite to ‘a’ measuring 26º, calculate the measure of ∠B opposite to ‘b’?

Solution:

Using the sine rule, we have sinA/a ​= sinB/b​ = sin26º/18​ = sin B/20​.

⇒ sin B = (9/10) ​sin26º or B ≈ 29.149º.

However, note that sin x = sin(180º – x). ∵ A + B < 180º and A + (180º – B) < 180º, another possible measure of B is approximately 180º – 29.149º = 150.851º.

Examples Using Law of Sines

What is Sine Formula?

Let us take an oblique triangle, i.e. a triangle with no right angle. Therefore, it is a triangle whose angles are all acute or a triangle with one obtuse angle. It is most useful for solving for missing information in a given triangle.

For example, if all three sides of the triangle are known, then Sine formula will allow us to find any or all of its three angles. Similarly, if two sides and the angle between these two sides is known, then the Sine formula allows us to find the third side length.

The Law of Sine

The Sine Rule is used in the following cases as follows:

• CASE-1: Given two angles and one side in triangle i.e. AAS or ASA.
• CASE-2: Given two sides and a non-included angle in triangle i.e. SSA.

The Sine Rule states that the sides of a triangle are in the proportional of the sines of the opposite angles. In form of mathematics:

Derivation of the Sine Formula

To derive the formula, erect an altitude through B and termed it ashB. Expressing hB in terms of the side and the sine of the angle will give the sine law formula.

To include angle B and side b in the above relationship, then construct an altitude through C and termed it as h_C.

Therefore, the sine function is the ratio of the side of the triangle opposite to angle and divided by the hypotenuse. Easy way to remember this ratio along with the ratios for the other trigonometric functions is with the mnemonic as SOH-CAH-TOA. Which are,

• SOH = Sine is Opposite over the Hypotenuse
• CAH = Cosine is Adjacent over the Hypotenuse
• TOA = Tangent is Opposite over the Adjacent

This ratio can be used to solve problems involving distance or height, or if you need to know an angle measure.

Solved Examples for the Sine Formula

Q.1. Solve triangle PQR in which ∠P=63.5^∘and∠Q=51.2^∘  and r = 6.3 cm.

Solution: First, calculate the third angle.

∠R=180–(63.5+51.2)=65.3^∘

Next, calculate the sides.

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