1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
stepladder [879]
3 years ago
15

Verify sine law by taking triangle in 4 quadrantExplain with figure.​

Mathematics
1 answer:
Ksivusya [100]3 years ago
7 0

Proof of the Law of Sines

The Law of Sines states that for any triangle ABC, with sides a,b,c (see below)

a

 sin  A

=

b

 sin  B

=

c

 sin  C

For more see Law of Sines.

Acute triangles

Draw the altitude h from the vertex A of the triangle

From the definition of the sine function

 sin  B =

h

c

    a n d        sin  C =

h

b

or

h = c  sin  B     a n d       h = b  sin  C

Since they are both equal to h

c  sin  B = b  sin  C

Dividing through by sinB and then sinC

c

 sin  C

=

b

 sin  B

Repeat the above, this time with the altitude drawn from point B

Using a similar method it can be shown that in this case

c

 sin  C

=

a

 sin  A

Combining (4) and (5) :

a

 sin  A

=

b

 sin  B

=

c

 sin  C

- Q.E.D

Obtuse Triangles

The proof above requires that we draw two altitudes of the triangle. In the case of obtuse triangles, two of the altitudes are outside the triangle, so we need a slightly different proof. It uses one interior altitude as above, but also one exterior altitude.

First the interior altitude. This is the same as the proof for acute triangles above.

Draw the altitude h from the vertex A of the triangle

 sin  B =

h

c

      a n d          sin  C =

h

b

or

h = c  sin  B       a n d         h = b  sin  C

Since they are both equal to h

c  sin  B = b  sin  C

Dividing through by sinB and then sinC

c

 sin  C

=

b

 sin  B

Draw the second altitude h from B. This requires extending the side b:

The angles BAC and BAK are supplementary, so the sine of both are the same.

(see Supplementary angles trig identities)

Angle A is BAC, so

 sin  A =

h

c

or

h = c  sin  A

In the larger triangle CBK

 sin  C =

h

a

or

h = a  sin  C

From (6) and (7) since they are both equal to h

c  sin  A = a  sin  C

Dividing through by sinA then sinC:

a

 sin  A

=

c

 sin  C

Combining (4) and (9):

a

 sin  A

=

b

 sin  B

=

c

 sin  C

You might be interested in
Can someone help me do 13, 14 and 15 thank u
sweet [91]

Answer:

Step-by-step explanation:

8 0
2 years ago
Write the polynomials in standard form:<br><br> (15–7xy^2)–(y^3–y^2x–15)
stepan [7]
<h3>Answer:</h3>

-y³ -6y²x +30

<h3>Step-by-step explanation:</h3>

When parentheses are eliminated and the y variable placed first in each term, the expression simplifies to ...

... 15 -7y²x -y³ +y²x +15

Arranging like terms together, we have ...

... -y³ +y²x(-7 +1) +(15 +15)

... -y³ -6y²x +30

_____

The two different 3rd-degree terms can be expressed in either order. I have chosen to list the term with the largest exponent first. The expression could also be written with the same-degree terms in "lexicographical order", meaning the expression would be written ...

... -6xy² -y³ +30

7 0
3 years ago
Which statement is best represented by the inequality c greater-than 3 and one-half?
luda_lava [24]

Answer: Chef Andre used more than 3 and one-half cups of flour.

Step-by-step explanation:

This is the > ‘greater than sign. For the question, we are to get the inequality that shows that"c is greater-than 3 and one-half".

Let Chef Andre be represented by c. Greater than is used to denote a situation when something is."more than" another thing. In this scenario,

Chef Andre used more than 3 and one-half cups of flour. This can be represented as C > 3 1/2.

This is the correct answer.

3 0
3 years ago
Read 2 more answers
I need help with a prople on my math for 8 grade it is y = -8x - 2
Nadusha1986 [10]

Answer:

M=8 Y=-2

Step-by-step explanation:

this is the slope and b

8 0
3 years ago
Help! Plz thank you!!!!
Brums [2.3K]

Answer:

i do not see a picture or an image of what you need.

Step-by-step explanation:

6 0
3 years ago
Other questions:
  • Norman weighed his loaded truck of apples at a weigh station. It weighed 15 tons 100 lbs. His empty truck weighed 3 tons 200 lbs
    12·2 answers
  • Select the property of equality used to arrive at the conclusion.
    9·1 answer
  • Explain how to find the exact value of cot 5pi/3 , including quadrant location.
    12·1 answer
  • Find a coefficient of x in -7x, and a coefficient of 9/4y.
    12·2 answers
  • Need help like now smart people plz ASAP
    12·1 answer
  • What is the greatest common factor (GCF) of 16 and 24?<br> GCF =
    8·2 answers
  • I am a cube that has a lenght, width, and hight of 1 unit
    13·1 answer
  • Elizabeth wants to ride her bicycle 28.6 miles this week. She has already ridden 19 miles. If she rides for 3 more days, write a
    12·1 answer
  • Someone please tell if this is right!
    12·1 answer
  • In 2005, there were 12,000 students at Beachon High. In 2010, there were 12,250. What is the rate of change un the number of stu
    9·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!