![Page 1: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/1.jpg)
Trigonometry: Laws of Sine and Cosine
Honors Precalculus
Mr. Velazquez
![Page 2: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/2.jpg)
Law of Sines: Recap
![Page 3: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/3.jpg)
Find the measure of angle S
Law of Sines: Practice
![Page 4: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/4.jpg)
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
![Page 5: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/5.jpg)
Find the perimeter of the triangle below, to the nearest whole unit.
Law of Sines: Practice
![Page 6: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/6.jpg)
The Ambiguous Case (SSA)
![Page 7: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/7.jpg)
Find the measure of angle B and the length of side b:
The Ambiguous Case (SSA)
a=20”
810
c=12”
![Page 8: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/8.jpg)
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
![Page 9: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/9.jpg)
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.
![Page 10: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/10.jpg)
Law of Cosines: SAS Triangles
![Page 11: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/11.jpg)
Law of Cosines: SAS Triangles
Find the length of side c and both missing angles:
![Page 12: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/12.jpg)
Law of Cosines: SSS Triangles
![Page 13: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/13.jpg)
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
![Page 14: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/14.jpg)
Finding Areas of Oblique Triangles
![Page 15: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/15.jpg)
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
![Page 16: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/16.jpg)
Finding Area Using Sine
Find the area of the given triangle:
071
b=5”
c=6”
![Page 17: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/17.jpg)
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
![Page 18: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/18.jpg)
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
![Page 19: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/19.jpg)
Applying the Laws of Sine and Cosine
![Page 20: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/20.jpg)
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.
![Page 21: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/21.jpg)
Applying Laws of Sine and Cosine
Find the angle 𝜃 between paces for both the herbivore and the carnivore.
![Page 22: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/22.jpg)
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
![Page 23: Trigonometry: Laws of Sine and Cosine...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](https://reader036.vdocument.in/reader036/viewer/2022070211/6100633e58a5d45f39287337/html5/thumbnails/23.jpg)
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)