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
Dahasolnce [82]
3 years ago
9

For any triangle ABC note down the sine and cos theorems ( sinA/a= sinB/b etc..)

Mathematics
1 answer:
SCORPION-xisa [38]3 years ago
8 0

Answer:

Step-by-step explanation:

Law of sines is:

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

Law of cosines is:

c² = a² + b² − 2ab cos C

Note that a, b, and c are interchangeable, so long as the angle in the cosine corresponds to the side on the left of the equation (for example, angle C is opposite of side c).

Also, angles of a triangle add up to 180° or π.

(i) sin(B−C) / sin(B+C)

Since A+B+C = π, B+C = π−A:

sin(B−C) / sin(π−A)

Using angle shift property:

sin(B−C) / sin A

Using angle sum/difference identity:

(sin B cos C − cos B sin C) / sin A

Distribute:

(sin B cos C) / sin A − (cos B sin C) / sin A

From law of sines, sin B / sin A = b / a, and sin C / sin A = c / a.

(b/a) cos C − (c/a) cos B

From law of cosines:

c² = a² + b² − 2ab cos C

(c/a)² = 1 + (b/a)² − 2(b/a) cos C

2(b/a) cos C = 1 + (b/a)² − (c/a)²

(b/a) cos C = ½ + ½ (b/a)² − ½ (c/a)²

Similarly:

b² = a² + c² − 2ac cos B

(b/a)² = 1 + (c/a)² − 2(c/a) cos B

2(c/a) cos B = 1 + (c/a)² − (b/a)²

(c/a) cos B = ½ + ½ (c/a)² − ½ (b/a)²

Substituting:

[ ½ + ½ (b/a)² − ½ (c/a)² ] − [ ½ + ½ (c/a)² − ½ (b/a)² ]

½ + ½ (b/a)² − ½ (c/a)² − ½ − ½ (c/a)² + ½ (b/a)²

(b/a)² − (c/a)²

(b² − c²) / a²

(ii) a (cos B + cos C)

a cos B + a cos C

From law of cosines, we know:

b² = a² + c² − 2ac cos B

2ac cos B = a² + c² − b²

a cos B = 1/(2c) (a² + c² − b²)

Similarly:

c² = a² + b² − 2ab cos C

2ab cos C = a² + b² − c²

a cos C = 1/(2b) (a² + b² − c²)

Substituting:

1/(2c) (a² + c² − b²) + 1/(2b) (a² + b² − c²)

Common denominator:

1/(2bc) (a²b + bc² − b³) + 1/(2bc) (a²c + b²c − c³)

1/(2bc) (a²b + bc² − b³ + a²c + b²c − c³)

Rearrange:

1/(2bc) [a²b + a²c + bc² + b²c − (b³ + c³)]

Factor (use sum of cubes):

1/(2bc) [a² (b + c) + bc (b + c) − (b + c)(b² − bc + c²)]

(b + c)/(2bc) [a² + bc − (b² − bc + c²)]

(b + c)/(2bc) (a² + bc − b² + bc − c²)

(b + c)/(2bc) (2bc + a² − b² − c²)

Distribute:

(b + c)/(2bc) (2bc) + (b + c)/(2bc) (a² − b² − c²)

(b + c) + (b + c)/(2bc) (a² − b² − c²)

From law of cosines, we know:

a² = b² + c² − 2bc cos A

2bc cos A = b² + c² − a²

cos A = (b² + c² − a²) / (2bc)

-cos A = (a² − b² − c²) / (2bc)

Substituting:

(b + c) + (b + c)(-cos A)

(b + c)(1 − cos A)

From half angle formula, we can rewrite this as:

2(b + c) sin²(A/2)

(iii) (b + c) cos A + (a + c) cos B + (a + b) cos C

From law of cosines, we know:

cos A = (b² + c² − a²) / (2bc)

cos B = (a² + c² − b²) / (2ac)

cos C = (a² + b² − c²) / (2ab)

Substituting:

(b + c) (b² + c² − a²) / (2bc) + (a + c) (a² + c² − b²) / (2ac) + (a + b) (a² + b² − c²) / (2ab)

Common denominator:

(ab + ac) (b² + c² − a²) / (2abc) + (ab + bc) (a² + c² − b²) / (2abc) + (ac + bc) (a² + b² − c²) / (2abc)

[(ab + ac) (b² + c² − a²) + (ab + bc) (a² + c² − b²) + (ac + bc) (a² + b² − c²)] / (2abc)

We have to distribute, which is messy.  To keep things neat, let's do this one at a time.  First, let's look at the a² terms.

-a² (ab + ac) + a² (ab + bc) + a² (ac + bc)

a² (-ab − ac + ab + bc + ac + bc)

2a²bc

Repeating for the b² terms:

b² (ab + ac) − b² (ab + bc) + b² (ac + bc)

b² (ab + ac − ab − bc + ac + bc)

2ab²c

And the c² terms:

c² (ab + ac) + c² (ab + bc) − c² (ac + bc)

c² (ab + ac + ab + bc − ac − bc)

2abc²

Substituting:

(2a²bc + 2ab²c + 2abc²) / (2abc)

2abc (a + b + c) / (2abc)

a + b + c

You might be interested in
Find the measurement of the angle indicated for each trapezoid
suter [353]

Answer:

45

Step-by-step explanation:

5 0
3 years ago
A Grocery store sells steak for $6.10 per pound.What would be the cost of 2 3/5 LB of steak?
77julia77 [94]
<h2>Answer:</h2>

<u>The cost would be </u><u>$15.86</u>

<h2>Step-by-step explanation:</h2>

The cost of 1 steak is $6.10

The cost of 2 3/5 steaks will be

Cost of  2 3/5 steaks = 6.10 * 2 3/5

Cost of  2 3/5 steaks = $15.86

8 0
3 years ago
Look at the example problem. Explain how you know 1 5/8 is little greater than 1 1/2.
natita [175]

Convert the fractions so both fractions have the same denominator. 1 1/2 is equal to 1 4/8 (1/2 * 4/4= 4/8). 1 4/8 is smaller than 1 5/8.

8 0
3 years ago
Find the value of x.
UNO [17]

Answer:

x = 20

Step-by-step explanation:

Intersecting Chords Theorem: ab = cd

Step 1: Label our variables

<em>a</em> = x

<em>b</em> = x - 11

<em>c</em> = x - 8

<em>d</em> = x - 5

Step 2: Plug into theorem

x(x - 11) = (x - 5)(x - 8)

Step 3: Solve for <em>x</em>

x² - 11x = x² - 8x - 5x + 40

x² - 11x = x² - 13x + 40

-11x = -13x + 40

2x = 40

x = 20

5 0
3 years ago
Read 2 more answers
Please help i don't understand. will mark brainliest. answer the both questions
vladimir1956 [14]

Answer:

1-Undefined

2-Positive

Step-by-step explanation:

5 0
2 years ago
Read 2 more answers
Other questions:
  • Use the stem-and-leaf plot of Monthly Sales Goals to answer the question that follows.
    11·1 answer
  • Slove for the variable 2(y - 5) = 5y + 23
    11·1 answer
  • A rule for creating a pattern is given in equation form below.
    15·2 answers
  • What is the length of QO?
    5·1 answer
  • Predict the number of chocolate chips in nine pancakes
    10·1 answer
  • there were 60 people in the town's first annual parade. That number has been growing by about 5% each year. predict the number o
    5·1 answer
  • True or False: y = x2 + 3 is a function.<br> A. True<br> B. False
    15·2 answers
  • Which operation will isolate the variable?
    15·2 answers
  • A math professor notices that scores from a recent exam are normally distributed with a mean of 61 and a standard deviation of 8
    9·1 answer
  • Write the equation of a line that<br> goes through the points (4,4) and<br> (-1,9).
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!