law of sines assignment active

The Law of Sines

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

a sin A = b sin B = c sin C

It works for any triangle:

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

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

a sin A = 8 sin(62.2°) = 8 0.885... = 9.04...

b sin B = 5 sin(33.5°) = 5 0.552... = 9.06...

c sin C = 9 sin(84.3°) = 9 0.995... = 9.04...

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

So now you can see that:

Is This Magic?

Not really, look at this general triangle and imagine it is two right-angled triangles sharing the side h :

The sine of an angle is the opposite divided by the hypotenuse, so:

a sin(B) and b sin(A) both equal h , so we get:

a sin(B) = b sin(A)

Which can be rearranged to:

a sin A = b sin B

We can follow similar steps to include c/sin(C)

How Do We Use It?

Let us see an example:

Example: Calculate side "c"

Now we use our algebra skills to rearrange and solve:

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):

sin A a = sin B b = sin C c

Example: Calculate angle B

Sometimes there are two answers .

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

Two possible answers.

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?"

Example: Calculate angle R

The first thing to notice is that this triangle has different labels: PQR instead of ABC. But that's OK. We just use P,Q and R instead of A, B and C in The Law of Sines.

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 there are two possible answers for R: 67.1° and 112.9° :

Both are possible! Each one has the 39° angle, and sides of 41 and 28.

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 (like below) and there is one solution

For example this triangle from before.

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

So this has only one solution.

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

Chapter 11: Functions

11.8 Sine and Cosine Laws

Right angle trigonometry is generally limited to triangles that contain a right angle. It is possible to use trigonometry with non-right triangles using two laws: the sine law and the cosine law.

The Law of Sines

The sine law is a ratio of sines and opposite sides. The law takes the following form:

[latex]\dfrac{a}{\text{sin }A}\hspace{0.25in} =\hspace{0.25in} \dfrac{b}{\text{sin }B}\hspace{0.25in} =\hspace{0.25in} \dfrac{c}{\text{sin }C}[/latex]

Sometimes, it is written and used as the reciprocal of the above:

[latex]\dfrac{\text{sin }A}{a}\hspace{0.25in} =\hspace{0.25in} \dfrac{\text{sin }B}{b}\hspace{0.25in} =\hspace{0.25in} \dfrac{\text{sin }C}{c}[/latex]

The law of sine is used when either two sides and one opposite angle of one of the sides are known, or when there are two angles and one side of one of the angles. If there are two given angles of a triangle, then all three angles are known, since [latex]A^{\circ} + B^{\circ} + C^{\circ} = 180^{\circ}.[/latex]

The sine law is a very useful law with one caveat in that it is possible to sometimes have two triangles (one larger and one smaller) that generate the same result. This is termed the ambiguous case and is described later in this section.

There are also textbook errors where the data given for the triangle is impossible to create. For instance:

Example 11.8.1

Can the following triangle exist?

If this triangle can exist, then the ratio of sines for the angles to the opposite sides should equate.

[latex]\dfrac{6}{\text{sin }30^{\circ}}\hspace{0.25in} = \hspace{0.25in} \dfrac{6}{\text{sin }30^{\circ}}\hspace{0.25in} = \hspace{0.25in} \dfrac{10}{\text{sin }120^{\circ}}[/latex]

Reducing this yields:

[latex]\dfrac{6}{0.5} \hspace{0.25in} = \hspace{0.25in}\dfrac{6}{0.5} \hspace{0.25in} = \hspace{0.25in} \dfrac{10}{0.866}[/latex]

In checking this out, we find that 12 = 12 ≠ 11.55.

This means that this triangle cannot exist.

Example 11.8.2

Find the correct length of the side opposite 120° in the triangle shown below.

For this triangle, the ratio to solve is:

[latex]\dfrac{6}{\text{sin }30^{\circ}}\hspace{0.25in}=\hspace{0.25in}\dfrac{6}{\text{sin }30^{\circ}} \hspace{0.25in}=\hspace{0.25in}\dfrac{x}{\text{sin }120^{\circ}}[/latex]

We only need to use one portion of this, so:

[latex]\dfrac{6}{\text{sin }30^{\circ}}\hspace{0.25in}=\hspace{0.25in}\dfrac{x}{\text{sin }120^{\circ}}[/latex]

Multiplying both sides of this by sin 120°, we are left with:

[latex]x=\dfrac{6\text{ sin }120^{\circ}}{\text{sin }30^{\circ}}[/latex]

This leaves us with [latex]x = 10.29[/latex].

Example 11.8.3

[latex]\dfrac{a}{\text{sin }44^{\circ}}\hspace{0.25in}=\hspace{0.25in}\dfrac{110}{\text{sin }86^{\circ}}[/latex]

Multiplying both sides by sin 44° leaves us with:

[latex]a=\dfrac{110\text{ sin }44^{\circ}}{\text{sin }86^{\circ}}[/latex]

Example 11.8.4

Find the unknown angle shown in the triangle shown below.

[latex]\dfrac{14}{\text{sin }A}\hspace{0.25in}=\hspace{0.25in}\dfrac{16}{\text{sin }52^{\circ}}[/latex]

Isolating sin A yields:

[latex]\text{sin }A=\dfrac{14\text{ sin }52^{\circ}}{16}[/latex]

We now need to take the inverse sin of both sides to solve for A:

[latex]\begin{array}{l} A=\text{sin}^{-1}\left(\dfrac{14\text{ sin }52^{\circ}}{16}\right) \\ \\ A=43.6^{\circ} \end{array}[/latex]

The Ambiguous Case

It is possible, when given the right data, to create two different triangles.

You can see from the triangle shown above that it is possible to have two angles, B 1 and B 2 , for side [latex]b[/latex]. Using the sine law, you will always end up solving for B 1 , the angle for the largest triangle. If you are trying to solve for the smaller triangle, then you only need to subtract B 1 from 180°.

For example, if B 1 = 50°, then B 2 = 180° − B 1 . This means B 2 = 180° − 50° or 130°.

If the angle you solve for when using the sine law is smaller than it should be, then correct for it as we just did above.

The Law of Cosines

The Law of Cosines is the generalized law of the Pythagorean Theorem [latex](a^2 + b^2 = c^2).[/latex]

The Law of Cosines is generally written in three different forms, which are as follows:

[latex]\begin{array}{l} a^2 = b^2 + c^2 - 2bc\text{ cos }A \\ b^2 = a^2 + c^2 - 2ac\text{ cos }B \\ c^2 = a^2 + b^2 - 2ab\text{ cos }C \end{array}[/latex]

All forms revert back to one of the three regular forms of the Pythagorean theorem [latex](a^2 = b^2 + c^2, b^2 = a^2 + c^2, c^2 = a^2 + b^2)[/latex] if [latex]A, B[/latex] or [latex]C[/latex] is 90°, since [latex]\text{cos }90^{\circ} = 0.[/latex] The following examples illustrate the usage of the cosine law in trigonometry

Example 11.8.5

For this triangle:

[latex]\begin{array}{l} A=\text{find} \\ \\ a=90 \\ \\ b=50 \\ \\ c=100 \end{array}[/latex]

Note that [latex]b[/latex] and [latex]c[/latex] could be switched around. Now, to solve:

[latex]\begin{array}{rrlllll} a^2 &= &\phantom{-}b^2 &+ &c^2 &-& 2bc\text{ cos }A \\ \\ 90^2 &= &\phantom{-}50^2 &+ &100^2 &- &2(50)(100)\text{ cos } A \\ 8100& =&\phantom{-}2500 &+ &10000& - &10000\text{ cos }A \\ -2500&&-2500&-&10000&& \\ \hline -4400& =&-10000&\text{ cos }&A&& \\ \\ \text{ cos }A& =&\dfrac{-4400}{-10000}&&&& \\ \\ B& =&\text{cos}^{-1}0.44&&&& \\ \\ B&=&63.9^{\circ}&&&& \end{array}[/latex]

Example 11.8.6

Find the unknown side shown in the triangle shown below.

[latex]\begin{array}{l} B=70^{\circ} \\ \\ a=95 \\ \\ b=\text{find} \\ \\ c=60 \end{array}[/latex]

[latex]\begin{array}{l} b^2=a^2+c^2-2ac\text{ cos }B \\ \\ b^2=95^2+60^2-2(95)(60)\text{ cos }70^{\circ} \\ b^2=9025+3600-11400(0.34202) \\ b^2=12625 - 3899 \\ b^2=8726 \\ \\ b=\sqrt{8726} \\ b=93.4 \end{array}[/latex]

Unlike with the law of sines, there should be no ambiguous cases with the law of cosines.

Solve all unknowns in the following non-right triangles using either the law of sines or cosines.

Answer Key 11.8

Intermediate Algebra Copyright © 2020 by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

Share This Book

Law of Sines

The law of sines establishes the relationship between the sides and angles of an oblique triangle(non-right triangle). Law of sines and law of cosines in trigonometry are important rules used for "solving a triangle". According to the sine rule, the ratios of the side lengths of a triangle to the sine of their respective opposite angles are equal. Let us understand the sine law formula and its proof using solved examples in the following sections.

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

• 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.

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,c 2 = a 2 + b 2 - 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. \( \frac{a - b}{a + b} = \frac{tan\frac{(A - B)}{2}}{tan\frac{(A + B)}{2}}\)

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'?

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.

Think out of the box:

Find the angles of a triangle if the sides are 12 units, 8.5 units, and 7.2 units respectively.

Examples Using Law of Sines

Example 1: Two angles and an included side is∠A = 47º and ∠B = 78º and c = 12.6 units. Find the value of a.

Given: ∠A = 47º and ∠B = 78º

∠A + ∠B + ∠C = 180º ⇒ 47º + 78º + ∠C = 180º ⇒ 125º + ∠C = 180º ⇒ ∠C = 180º - 125º ⇒ ∠C = 55º

We shall apply the sine law to find the side of the triangle.

a/sin A = c/sin C

⇒ a/sin 47º = 12.6/sin 55º

Answer: a = 11.24 units

Example 2: It is given ∠A = 47º, ∠B = 78º, and the side c = 6.3. Find the length a.

To find: Length of a

∠A = 47º, ∠B = 78º, and c = 6.3.

Since, the sum of all the interior angles of the triangle is 180 ∘,

∠A + ∠B + ∠C=180º ⇒ 47º + 78º + ∠C = 180º ⇒ ∠C = 55º

Using law of sines formula,

a/sinA = b/sinB = c/sinC ⇒ a/sinA = c/sinC ⇒ a/sin47º = 6.3 / sin55º ⇒ a = 6.3 / sin55º × sin47º ⇒ a = 5.6

Answer: a = 5.6

Example 3: For a triangle, it is given a = 10 units c = 12.5 units and angle C = 42º. Find the angle A of the triangle.

To find: Angle A

a = 10, c = 12.5, and angle C = 42º.

⇒ a/sinA = b/sinB = c/sinC ⇒ 10/sinA = 12.5/sin 42º ⇒ sin A = 0.5353 ⇒ ∠A = 32.36º

go to slide go to slide go to slide

Book a Free Trial Class

Practice Questions on Law of Sines

go to slide go to slide

FAQs on Sine Law

What is meant by law of sines.

The Law of sines gives a relationship between the sides and angles of a triangle. The law of sines in Trigonometry can be given as, a/sinA = b/sinB = c/sinC, where, a, b, c are the lengths of the sides of the triangle and A, B, and C are their respective opposite angles of the triangle.

When Can We Use Sine Law?

Sine law finds application in solving a triangle, which means to find the missing angle or side of a triangle using the requisite given data. We can use the sine law to find,

• Side of a triangle
• The angle of a triangle
• Area of a triangle

What is the Sine Rule Formula?

The sine rule formula gives the ratio of the sides and angles of a triangle. The sine rule can be explained using the expression, a/sinA = b/sinB = c/sinC. Here a, b, c are the length of the sides of the triangle, and A, B, C are the angles of the triangle.

What are the Different Ways to Represent Sine Rule Formula?

Sine law can be represented in the following three ways. These three forms are as given below,

In Which Cases Can We Use the Sine Law?

The sine law can be used for three purposes as mentioned below,

• To find the length of sides of a triangle
• To find the angles of the triangle
• To find the area of the triangle

Can Sine Law be Used on a Right Triangle?

The sine law can also be used for a right triangle. sine law can be used in oblique(non-right) as well as in a right triangle to establish a relationship between the ratios of sides and their respective opposite angles.

What are the Possible Criteria for Law of Sines?

The criteria to use the sine law is to have one of the following sets of data known to us,

• A pair of lengths of two sides of a triangle and an angle.
• A pair of angles of a triangle and the length of one side.

Does Law of Sines Work With 90 Degrees?

Yes, the law of sines and the law of cosines can be applied to both the right triangle and oblique triangle or scalene triangle to solve the given triangle.

What is the Law of Sines Ambiguous Case?

The law of sines ambiguous case is the case that occurs when there can be two possible solutions while solving a triangle. Given a general triangle, the following given conditions would need to be fulfilled for the ambiguous case,

• Only information given is the angle A and the sides a and c.
• Angle A is acute, i.e., ∠A < 90°.
• Side a is shorter than side c ,i.e., a < c.
• Side a is longer than the altitude h from angle B, where h = c sin A, .i.e., a > h.

What are the Applications of the Law of Sines?

The law of sines can be applied to find the missing side and angle of a triangle given the other parameters. To apply the sine rule, we need to know either two angles and one side of the triangle (AAS or ASA) or two sides and an angle opposite one of them (SSA).

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

Chapter 3: Laws of Sines and Cosines

• Last updated
• Save as PDF
• Page ID 112392

• Katherine Yoshiwara
• Los Angeles Pierce College

The first science developed by humans is probably astronomy. Before the invention of clocks and calendars, early people looked to the night sky to help them keep track of time. What is the best time to plant crops, and when will they ripen? On what day exactly do important religious festivals fall?

By tracking the motions of the stars, early astronomers could identify the summer and winter solstices and the equinoxes. The rising and setting of certain stars marked the hours of the night.

If we think of the stars as traveling on a dome above the Earth, we create the celestial sphere . Actually, of course, the Earth itself rotates among the stars, but for calculating the motions of heavenly objects, this model works very well.

Babylonian astronomers kept detailed records on the motion of the planets, and were able to predict solar and lunar eclipses. All of this required familiarity with angular distances measured on the celestial sphere.

To find angles and distances on this imaginary sphere, astronomers invented techniques that are now part of spherical trigonometry . The laws of sines and cosines were first stated in this context, in a slightly different form than the laws for plane trigonometry.

On a sphere, a great-circle lies in a plane passing through the sphere’s center. It gives the shortest distance between any two points on a sphere, and is the analogue of a straight line on a plane. A spherical angle is formed where two such arcs intersect, and a spherical triangle is made up of three arcs of great circles.

The spherical law of sines was first introduced in Europe in 1464 by Johann Muller, also known as Regiomontus, who wrote:

"You, who wish to study great and wondrous things, who wonder about the movement of the stars, must read these theorems about triangles. ... For no one can bypass the science of triangles and reach a satisfying knowledge of the stars.”

• 3.0: Obtuse Angles
• 3.1: The Law of Sines
• 3.2: The Law of Cosines
• 3.3: Chapter 3 Summary and Review
• 3.4: Chapter 3 Activity

The Laws of Cosines and Sines

The law of sines, the law of cosines.