Topic 2 The Sine Law Unit 3 Topic 2. Before We Start.

Topic 2The Sine Law

Unit 3 Topic 2

Before We Start

• Sometimes we must work with triangles that are not right angle triangles. A triangle that does not contain a right angle is called an oblique triangle.

• The Pythagorean Theorem () and the trigonometric ratios (SOH CAH TOA) cannot be used with oblique triangles.

• Instead, we use the Sine Law or the Cosine Law.

Explore• We are going to develop the Sine Law using the

oblique triangle below:

▫ 1) Draw in the height of the triangle and label it h.▫ 2) Using the two right triangles formed, write a

trigonometric ratio for sin A and for sin C.▫ 3) Using the ratios from step 2, isolate for h.▫ 4) Since both equations are equal to h, equate

them to eliminate h.▫ 5) Divide both sides of the equation by ac.

A C

B

c

b

a

You should get…• When using the steps on the previous slide, you

should have gotten the following:

A C

B

c

b

a

sinh

Ac

sinh

Ca

sinh a Csinh c A

sin sinc A a C

sin sinc A a C

ac ac

sin sinA C

a c

Information• The Sine Law is a relationship between the

sides and angles in an oblique triangle.

C

ab

cA B

Example 1Determining the length

In DEF, calculate the length of d to the nearest tenth.

a) b)

Try this on your own first!!!!

Example 1a: Solution

Before we calculate anything, we must identify a given angle across from a given side. This forms our ‘complete set.’

Next we identify the side we are solving for and the given angle across from it. This forms our ‘incomplete set.’

dsin 78 sin 42

5.4 d

sin 78 5.4sin 42d

5.4sin 42

sin 78d

3.7d m

Set up the equation.

Cross multiply.

Divide.

Example 1b: Solution

Before we calculate anything, we must identify a given angle across from a given side. This forms our ‘complete set.’

Next we identify the side we are solving for and the given angle across from it. This forms our ‘incomplete set.’d

sin 38 sin115

9.8 d

sin 38 9.8sin115d

9.8sin115

sin 38d

14.4d m

Set up the equation.

Cross multiply.

Divide.

In this question, we don’t have the angle across from the side we are solving for. We can find it using the triangle sum theory.180° -27° -38° = 115°

115°

Example 2Determining the angle

In ABC, calculate the measure of to the nearest degree.

a) b)

Try this on your own first!!!!

Example 2a: Solution

Before we calculate anything, we must identify a given angle across from a given side. This forms our ‘complete set.’

Next we identify the angle we are solving for and the given side across from it. This forms our ‘incomplete set.’

sin120 sin

11.8 7.6

A

7.6sin120 11.8sin A

7.6sin120sin

11.8A

Set up the equation.

Cross multiply.

Divide.

Use the inverse of sine to solve for the angle.

1 7.6sin120sin 3711.8

Example 2b: Solution

Before we calculate anything, we must identify a given angle across from a given side. This forms our ‘complete set.’

Next we identify the angle we are solving for and the given side across from it. This forms our ‘incomplete set.’

sin 43 sin

12.2 16.2

A

16.2sin 43 12.2sin A

16.2sin 43sin

12.2A

Set up the equation.

Cross multiply.

Divide.

Use the inverse of sine to solve for the angle.

1 16.2sin 43sin 6512.2

Example 3Determining the lengths and angle

James is building a greenhouse. To take advantage of the sunshine, James constructs the roof as illustrated.    a) Determine the measure of , to the nearest

degree.   b) Determine the lengths of the roof sections AC

and BC, to the nearest tenth of a metre.

Try this on your own first!!!!

We can find easily, since we have the other 2 angles. 180° - 30° - 40° = 110°

Example 3: Solution 110°

Side AC Side BC

sin110 sin 30

7 a

Identify the complete set and the incomplete step to set up the equation for each.

Cross multiply.

Divide.

sin110 sin 40

7 b

sin110 7sin 40b sin110 7sin30a

7sin30

sin110a

7sin 40

sin110b

4.8b m 3.7a m

Example 4Determining the length given two triangles

Calculate the height, h, of the cliff given the following diagram, to the nearest metre.

Try this on your own first!!!!

In order to solve for h in the right-angled triangle, I first need the side common to both triangles.

I need to use sine law to solve for the red side. I need to use the third (missing) angle to come up with my complete ratio.

66°sin 66 sin 49

185 a

a

sin 66 185sin 49a

185sin 49

sin 66a

152.83451a m

Example 4: Solution (continued)

h

3165

185 m

49

Now we have the known side that we need in order to find h.

We can use the first triangle, and label it according to the reference angle 31 .⁰

152.8m

Since we have the opposite side and the adjacent side, we can use the tan ratio.

66

tan

tan31152.83451

152.83451tan(31 )

92

oppositeadjacent

hm

h

h m

opposite

adjacent

Need to Know:

• A triangle that does not contain a right angle is called an oblique triangle.

• When solving for unknown values in an oblique triangle, the Pythagorean Theorem and SOH CAH TOA cannot be used.

Need to Know:

• If given a side length and the angle opposite to it, the Sine Law can be used to find the missing side length or angle.

• When finding a missing angle, use the inverse of sine (sin-1).

You’re ready! Try the homework from this section.