Law of Sines - Solving Oblique Triangle
Summary
TLDRIn 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
- π The video discusses the Law of Sines, a mathematical principle used to solve oblique triangles.
- π 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).
- π The script introduces an alternative version of the Law of Sines formula with sine functions in the numerator and sides in the denominator.
- π 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.
- π§ It explains that the sum of angles in a triangle is 180 degrees, and uses this to find the missing angle A.
- π’ The script demonstrates the use of a calculator to solve for side B using the Law of Sines formula.
- π 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.
- π The process is repeated to find side C, again using the Law of Sines and cross-multiplication.
- π The video emphasizes the importance of accurate calculations and rounding off to the appropriate number of decimal places.
- π’ The presenter, Smithy Turgon, encourages viewers to follow him on social media and subscribe to his channel for more educational content.
- π 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
π 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.
π 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.
π 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.
Mindmap
Keywords
π‘Law of Sines
π‘Oblique Triangle
π‘Trigonometry
π‘Formula
π‘Angle
π‘Side
π‘Calculator
π‘Cross Multiply
π‘Sine
π‘Degrees
π‘Example
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
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