Trigonometry: Laws of Sine and Cosine

Honors Precalculus

Mr. Velazquez

Law of Sines: Recap

Find the measure of angle S

Law of Sines: Practice

Points A and B are on one side of a river, 100 ft apart, with point C on the opposite side. The angles A and B measure 70° and 60° respectively. What is the distance from point A to point C, to the nearest foot?

Law of Sines: Practice

Find the perimeter of the triangle below, to the nearest whole unit.

Law of Sines: Practice

The Ambiguous Case (SSA)

Find the measure of angle B and the length of side b:

The Ambiguous Case (SSA)

a=20”

810

c=12”

Sketch and label the following triangle in your notebook and follow along with me.

Law of Cosines: Derivation

A B

C

a b

a cos B c – a cos B

h = a sin B

Law of Cosines

We can use Law of Cosines to solve for the missing sides and angles of any oblique triangle, IF we are given (a) one angle and the lengths of the two sides adjacent to that angle, or (b) the lengths of all three sides of the triangle.

Law of Cosines: SAS Triangles

Law of Cosines: SAS Triangles

Find the length of side c and both missing angles:

Law of Cosines: SSS Triangles

Three sides of a triangle measure 20 m, 30 m, and 40 m. Find the measure of all three angles of this triangle, to the nearest degree.

Law of Cosines: SSS Triangles

20 30

40

Finding Areas of Oblique Triangles

Because the area of a triangle is 𝐴 =1

2𝑏ℎ, we can use the

properties of trig functions to prove that the area of any triangle is one-half the product of the length of two sides times the sine of the angle between them. Or:

𝑨 =𝟏

𝟐𝒃𝒄 𝐬𝐢𝐧𝑨 =

𝟏

𝟐𝒂𝒃 𝐬𝐢𝐧𝑪 =

𝟏

𝟐𝒂𝒄 𝐬𝐢𝐧𝑩

Finding Area Using Sine

A B

C

a b

a cos B c – a cos B

h = a sin B

Finding Area Using Sine

Find the area of the given triangle:

071

b=5”

c=6”

The area of any triangle whose sides are of length 𝑎, 𝑏 and 𝑐 is:

𝑨 = 𝒔(𝒔 − 𝒂)(𝒔 − 𝒃)(𝒔 − 𝒄) Where 𝑠 is the semiperimeter of the triangle, meaning

𝑠 =1

2(𝑎 + 𝑏 + 𝑐).

Heron’s Formula for Area

Angelo’s triangular yard has the measurements of 55 feet, 80 feet and 95 feet. Find the area of Angelo’s yard, to the nearest square foot.

Heron’s Formula: Examples

Applying the Laws of Sine and Cosine

Applying Laws of Sine and Cosine

069

500 feet

058

A surveyor found the length of the riverbank and the angles of the two points on the opposite side of the river. Find the distance from point A to B.

Applying Laws of Sine and Cosine

Find the angle 𝜃 between paces for both the herbivore and the carnivore.

Applying Laws of Sine and Cosine

If you start on Island B, what bearing should you travel to reach Island C? What bearing should you travel to reach Island A?

2 Miles

3 Miles 4 Miles

Find all missing sides and angles for the two triangles shown.

Exit Ticket: Laws of Sine and Cosine

049

8 inchesb

7 inchesc

a 7 inchesa

8 inchesb

8 inchesc

Homework: Pg. 723-724, #1-21 (ALL)