Law of Sines - Solving Oblique Triangle

Summary

In this educational video, Smithy Turgon introduces the Law of Sines, a fundamental principle used for solving oblique triangles. He demonstrates the application of the Law of Sines formula by working through an example involving a triangle with given angles and a side length. The video guides viewers step-by-step to find the missing angles and sides, using a calculator to perform the necessary trigonometric calculations. Smithy Turgon emphasizes the importance of understanding the relationship between the angles and sides of a triangle, ultimately providing the measurements for all sides and angles in the example.

Takeaways

• 📚 The video discusses the Law of Sines, a mathematical principle used to solve oblique triangles.
• 🔍 The Law of Sines formula is presented as a ratio of the sides of a triangle to the sines of their opposite angles: a/sin(A) = b/sin(B) = c/sin(C).
• 📐 The script introduces an alternative version of the Law of Sines formula with sine functions in the numerator and sides in the denominator.
• 📈 The video provides an example problem involving a triangle with given angles B and C, and side a, aiming to find angle A and sides b and c.
• 🧭 It explains that the sum of angles in a triangle is 180 degrees, and uses this to find the missing angle A.
• 🔢 The script demonstrates the use of a calculator to solve for side B using the Law of Sines formula.
• 📝 Cross-multiplication is used to isolate the unknown side B in the ratio, and the sine of the known angle is used to find its length.
• 📉 The process is repeated to find side C, again using the Law of Sines and cross-multiplication.
• 📊 The video emphasizes the importance of accurate calculations and rounding off to the appropriate number of decimal places.
• 📢 The presenter, Smithy Turgon, encourages viewers to follow him on social media and subscribe to his channel for more educational content.
• 👋 The video concludes with a reminder of the presenter's name and a friendly sign-off.

Q & A

What is the law of sines used for?

The law of sines is used for solving oblique triangles, particularly when you have certain angles and sides and need to find the missing parts.

What is the formula for the law of sines?

The formula for the law of sines is \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \), where lower case letters represent the sides of the triangle and the corresponding upper case letters represent the angles opposite those sides.

In the given example, what are the known values in triangle ABC?

In the example, angle B is 141 degrees, angle C is 23 degrees, and the length of side a is 9 units.

How is the missing angle A calculated in the example?

Angle A is calculated using the formula \( 180^\circ - \text{Angle B} - \text{Angle C} \), which in this case is \( 180^\circ - 141^\circ - 23^\circ = 16^\circ \).

What is the first step to find the length of side B using the law of sines?

The first step is to use the formula \( \frac{a}{\sin A} = \frac{b}{\sin B} \) and cross-multiply to solve for side B.

How is the length of side B found in the example?

By cross-multiplying \( 9 \sin 141^\circ \) with \( \sin 16^\circ \) and dividing, the length of side B is approximately 20.5 units.

What formula is used to find the length of side C?

The same law of sines formula is used, but with the known values for side a and angle C to find the length of side C.

How is the length of side C calculated in the example?

By cross-multiplying \( 9 \sin 23^\circ \) with \( \sin 16^\circ \) and dividing, the length of side C is approximately 12.8 units.

What is the significance of the law of sines in solving for the missing sides of a triangle?

The law of sines allows you to relate the ratios of the sides of a triangle to the sines of their opposite angles, which is crucial when you have some angles and sides known and need to find the others.

What is the final step in the example after finding the missing angles and sides?

The final step is to verify that all the angles add up to 180 degrees and that the sides satisfy the law of sines, ensuring the solution is correct.

Outlines

00:00

📚 Introduction to the Law of Sines

Smithy Turgon introduces the Law of Sines in the context of solving oblique triangles. The formula is presented as a ratio of sides to the sines of their opposite angles: a/sinA = b/sinB = c/sinC. An alternative version of the formula is briefly mentioned. The first example involves a triangle with given angles B and C, and side a, aiming to find angle A, side b, and side c. The process of identifying the missing parts of the triangle is explained, emphasizing the relationship between angles and their opposite sides.

05:01

🔍 Calculating Missing Sides Using the Law of Sines

The video script details the process of using the Law of Sines to calculate the missing sides of the triangle. After determining the missing angle A to be 16 degrees by using the angle sum property of a triangle, the script moves on to find side B. It uses the Law of Sines formula 'a/sinA = b/sinB' and cross-multiplication to solve for side B, resulting in an approximate value of 20.5 units. The explanation includes the steps of cross-multiplying and dividing by the sine of the known angle to isolate the unknown side.

10:04

📐 Final Calculations for Side C and Conclusion

The script concludes with the calculation of the missing side C using a similar application of the Law of Sines as used for side B. The process involves cross-multiplying the known values and solving for side C, which is found to be approximately 12.8 units. The video ends with a summary of the learned concepts and an invitation for viewers to follow the channel and social media accounts for updates. The presenter, Teacher Gone, signs off with a farewell.

Keywords

💡Law of Sines

The Law of Sines is a fundamental principle in trigonometry used to solve for unknown angles and sides of a triangle, particularly oblique triangles. In the video, it is the central formula used to solve the given problem, with the formula presented as 'a/sin(a) = b/sin(b) = c/sin(c)'. The script demonstrates its application in finding the missing angle and sides of triangle ABC.

💡Oblique Triangle

An oblique triangle is a type of triangle that does not contain a right angle, meaning all angles are less than 90 degrees. The video's theme revolves around solving for unknown elements in an oblique triangle using the Law of Sines, with triangle ABC being the example where angles and sides are calculated.

💡Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. In the context of the video, trigonometry is essential for understanding and applying the Law of Sines to solve for the missing elements of triangle ABC.

💡Formula

A formula in mathematics is a concise way of expressing information symbolically, and in the video, the formula refers specifically to the Law of Sines. The script discusses two versions of the formula, one with sides over sines and another with sines over sides, to solve for the unknowns in the triangle.

💡Angle

An angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint. In the video, angles B and C are given, and angle A is calculated using the Law of Sines, illustrating the process of determining angles in a triangle.

💡Side

In the context of a triangle, a side refers to the line segment between two vertices. The script uses the term 'side' to denote the lengths of the sides of triangle ABC, with side a being given and sides b and c being calculated using the Law of Sines.

💡Calculator

A calculator is a device or software used to perform arithmetic operations. In the video, the term 'calculator' is used metaphorically to describe the process of performing the calculations needed to apply the Law of Sines and find the missing elements of the triangle.

💡Cross Multiply

Cross multiplication is a technique used in algebra to solve equations involving fractions. The video script describes using cross multiplication to solve for side B and side C of the triangle by equating the ratios formed by the Law of Sines.

💡Sine

The sine function is a fundamental trigonometric function that relates the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. In the video, sine is used in the context of the Law of Sines to calculate the unknown angles and sides of the oblique triangle ABC.

💡Degrees

Degrees are a unit of measurement for angles, with a full circle being 360 degrees. The video script provides the angles of triangle ABC in degrees and uses them in the Law of Sines formula to find the missing angle and sides.

💡Example

An example in an educational context is a specific case used to illustrate a concept or principle. The video provides a detailed example of triangle ABC to demonstrate the application of the Law of Sines in solving for unknown angles and sides.

Highlights

Introduction to the Law of Sines and its application in solving oblique triangles.

Explanation of the Law of Sines formula: a/sinA = b/sinB = c/sinC.

Understanding the relationship between sides (a, b, c) and angles (A, B, C) in a triangle.

Starting with an example problem involving a triangle with given angle B, angle C, and side a.

Listing the known and unknown parts of the triangle to set up the problem.

Using the sum of angles in a triangle to find the missing angle A.

Calculating angle A as 16 degrees using the Law of Sines.

Determination of the location of sides a, b, and c relative to the given angles.

Using the Law of Sines to solve for side B when angle A and side a are known.

Cross-multiplying to find the length of side B using the sine values.

Calculation of side B resulting in approximately 20.5 units.

Moving on to solve for side C using a similar method as for side B.

Cross-multiplying to find the length of side C with the known values.

Calculation of side C resulting in approximately 12.8 units.

Emphasizing the practical application of the Law of Sines in solving for unknowns in triangles.

Encouragement for viewers to learn from the video and follow the channel for more educational content.

Invitation to subscribe and engage with the content for updates on latest uploads.

Transcripts

hi guys it's Smithy turgon in today's

video we will talk about the law of

science the slow of science is

particularly used in solving oblique

triangles so without further ado

let's do this topic

so first before we solve a problem

for the law of science let us discuss

first what is the formula used for the

law of science what we have here is a

over sine a

is equal to B over sine B is equal to C

over sine C

now we have here a different version of

it we're in we will only flip the given

formula or the given fractions where in

design a sine B and sine C are in the

numerator while a B and C are in the

denominator

don't worry about it now let me discuss

first about this formula

small letter a small letter B small

letter C indicates or represents the

sides of an even oblique triangle while

this capital A B and C represents the

angles inside the given triangle so

let's start with this first example

in triangle ABC we are given angle B

which is 141 degrees this is angle B

angle C is 23 degrees

and a is 9 units the length of side a is

9 units

what we have here the problem is find

angle e or the measurement of angle a

side B and side C

uh

first we will try to list down all the

different parts or all the six parts of

a given triangle

I am starting with

dangos for the angles let's start with

Delta e

angle B

and angle C so in a given problem it is

already given that we have angle BS 141

degrees

and for angle C which is 23

degrees now

this will be the missing angle so we

will put a question mark here

for us to be reminded we need to solve

for a and next after the three angles we

will list down the three different sides

we have side a

side B

and side C now sir how can we determine

where's the location of the three

different sides Leon reference net and

are the given angles

here

is 9 units meaning this is side e

so what is the basis since this one is

angle e

your side a is opposite to your angle a

if this angle a automatically this is

now sure what about side B if this is

your angle B automatically this is your

side B

PSI angle C this is your side C so what

we have here is 9 for the value of a and

we are missing the value of side b or

the length of side B inside C let's get

started

for this part we will start with

foreign

we will use this casual calculator

where in we will solve first for angle e

and formula Net10 for angle e

is angle e

is equal to

180 degrees

minus

angle B

plus angle

C

your angle a

is still missing 180 degrees minus

you add the angle B plus angle C that is

141

Plus

23 degrees

so that is 100

64 degrees

minus 180 minus 164

our answer is 16 degrees meaning your

angle a

is 16 degrees that's easy as that so

this is 16 let me use another ink or

another color or the answer

16 degrees

which is side B and side C where it

because

for

side B

etoha

for side B

another

side B is we will check

okay out of these three formulas

since

angle a and side a meaning we will

definitely use

a over sine a

okay okay

to solve for

solve

for

side B

thing again

use this

you have e

over

sine e

is equal to because

we need to find B

B

over

sine b sir I don't know

um

instead of a over sine a Hindi

um

foreign

over your sine a sign

100 sorry 16 degrees

it's a call to your B

you have your Bia I just added paper B

your B is missing

over

sign your B is 141 degrees so we will

cross multiply

cross multiplying

semi missing variable so that would be

B

Iha sine

16 degrees is equal to

X across multiplying that and this one

and

and that would be

nine

sine

141 degrees so what's next

meaning we need to cancel this out

by dividing both sides by 60 sine 16

degrees

over

sine 16 degrees

cancel cancel

and as you can see

so we have here our B

which is equal to what is the value of B

so

guys

so that you will directly use the

calculator

so we have 9 sine 141 9 sine 141

okay

divided by

sine 16.

okay

so you can see Valentine's a good

20.5483161611

I'm just getting the one decimal place

value so I will stop with

20

.5

units

foreign

of

side bipero let me give you uh an exact

answer since we you know we rounded off

the

decimal again

approximately yeah

meaning your B is approximately

20.5 units

20.5 units

okay

that's approximately

now let's move on with the other missing

part which is letter c

song for C

solve

for

C

emetery a over sine e

a over

sine a is equal to this one

C over

sine C

this is nine

over

sine 16 degrees

is equal to your C is still missing

over sine

your C is 23 degrees

cross multiply guys

okay

meaning we have

C sine 16 degrees is equal to

cross multiply olita

okay this is nine

sine 23 degrees

we need to divide this

by sine 16

Trend by sine 16 degrees so we can

cancel this out

so as you can see we have C

tapos

nine

sine 23 degrees

divided by

sine 16 degrees so this is the answer

the answer is this one so round of

nothing

this is approximately

12.8

units

same guys

this one is 12.8 units

so I hope you guys should learn

something from this video and

if you want to follow me on my social

media accounts just subscribe to this

channel Facebook page

again guys if you're new to my channel

don't forget to like And subscribe

button hit the Bell button for you to be

updated latest uploads again it's me

teacher gone my name is

bye bye