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Right Triangles

Angles in Standard Position

Sine Law & Ambiguous Case

Cosine Law

Problem Solving

Summary&Test

 

jdsmathnotes

 

 


 UNIT 6 : BASIC TRIGONOMETRY WITH TRIANGLES

 LESSON 3: SINE LAW

 

 

The Sine Law:

The sine law is used to solve oblique triangles, that is triangles which are not right angled.

 

A

 

 

c b

 

 

 

B a C

 

 

Given two angles and a side (AAS case)

Example 1:

 

A

 

 

 

c b

 

 

 

B 48o 29o C

22.4 cm

 

Solution:

 

First find the measure of < A: 180o (48o + 29o) = 103o

Now find c using the sine law:

Now find b using the sine law:

 

 

Given two sides and an ACUTE angle (< 900) opposite one of the sides [the non-contained angle] (SSA)

Example 2:

Note: Here the longer side (23.1) is opposite the given acute angle < E is acute. There is only one triangle in this case.

D

 

 

 

16.4 cm 23.1 cm

 

 

 

E 56.4o F

Solution:

 

 

Now find < D: 180 (56.40 + 36.30) = 87.30

 

Next find d using the sin law:

 

 

Given two sides and an ACUTE angle where the shorter side (b) is opposite the given angle (< B) (SSA)

Example 3:

Solution: There are 3 possibilities here, one of which is the so-called ambiguous case where there are two triangles which satisfy the given conditions.

 

C

 

 

 

b a

 

 

 

A ( B

 

 

C

 

 

 

b h a

 

 

 

A ( B

D

 

C

 

 

 

b h b a

 

 

 

A1 ( B

D A2

 

 

 

C

 

 

 

9.6 h 9.6 12.4

 

 

 

A1 420 B

A2

 

Hence there are two triangles to solve here.

 

Triangle 1:

C

 

 

 

9.6 12.4

 

 

 

A1 420 B

 

 

 

 

Triangle 2 : Recall from above that < CA2B = 120.20

 

C

 

 

 

9.6 12.4

 

 

A2 120.20 420 B

 

 

 

 

 

 

C

 

b=h a

 

 

A B

 

 

 

 

C

 

 

 

b h b a

 

 

A A

( B

 

The SSA case where the given angle B is OBTUSE (> 900)

Example 4:

 

 

C

 

 

 

b a

 

 

 

 

A B

 

 

 

 

 

 

C

 

 

 

b a

 

 

 

 

A B

 

 

Example 4(a):

 

 

C

 

 

 

13.5 10.4

 

 

 

 

A B

1010

 

Solution: This is the SSA case with a given obtuse angle and the side opposite this angle is the longer side.

There will be only one triangle to solve.

First find < A using the sine law.

 

 

 

Text Box: Summary of main ideas:
	Use the sine law when given 2 angles and a side (AAS) or two sides and a non-contained 
      angle (SSA).
	For the SSA situation, where the given angle is acute; the ambiguous case occurs if the side opposite the given angle is the shorter of the two given sides. (ex. 3(a) above)
	To solve the triangle in this case, proceed as follows: 
1.	Calculate the height of the triangle (h = a sinB)
2.	If  b > h then there are two triangles
3.	If b = h there is one right triangle
4.	If b < h there is no triangle
	For the SSA situation, where the given angle is acute; if the side opposite the given angle is the longer side, there is one solution. (ex. 2 above)
	For the SSA situation, where the given angle is obtuse; if the side opposite the given angle is the longer side, there is one solution.  (ex. 4(a) above)
	For the SSA situation, where the given angle is obtuse; if the side opposite the given angle is the shorter side, there is no solution
	For the AAS situation, there is one solution.
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


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