1. LT 8.5 Use the law of sines and the law of cosines to solve triangles

2. All Triangles In this lesson, you will learn to solve any triangle.
To do so, you will need to calculate trigonometric ratios for angle measures up to 180°. You can use a calculator to find these values.

3. Example 1
Use a calculator to find each trigonometric ratio. Round to the nearest hundredth.
1a. tan175˚
1b cos92˚
1c. sin160˚
tan 175° = -.09
cos92° = -.03
sin160°= .34

4. Helpful Hint
Do not round your answer until the final step of the computation. If a problem has multiple steps, store the calculated answers to each part in your calculator.

5. Law of Cosines Helpful Hint
𝑜𝑝𝑝 2 = 𝑎𝑑𝑗 𝑎𝑑𝑗 (adj)(adj)cos(opp angle)

6. Example 4 – Use the law of cosines
Find the measure of XZ. Round to the nearest tenth.
𝑋𝑍 2 = (30)(35)cos110
𝑋𝑍 2 =
XZ = 53.3

7. Example 5 – Use the law of cosines
Find m<T. Round lengths to the nearest tenth and angle measures to the nearest degree.
Cos (opp angle) = (𝑜𝑝𝑝 2 − 𝑎𝑑𝑗 2 − 𝑎𝑑𝑗 2 ) (−2 𝑎𝑑𝑗 𝑎𝑑𝑗 )
Cos(T) = (7 2 − − ) (−2(13)(11))
CosT = − =
<T = 33°

8. Example 6 – sailing application
A sailing club has planned a triangular racecourse, as shown in the diagram. How long is the leg of the race along BC? How many degrees must competitors turn at point C? Round the length to the nearest tenth and the angle measure to the nearest degree.
2.8 miles and 83º

9. End of Day 1

10. Law of Sines

11. Example 2 – Use the law of sines
Find the measure of FG. Round to the nearest tenth.
FG = 33.7

12. Example 3 – Use the law of sines
Find m<Q. Round lengths to the nearest tenth and angle measures to the nearest degree.
<Q = 36º

13. Example – Use the law of sines
Round lengths to the nearest tenth and angle measures to the nearest degree.
PR = 10.3
<Q = 93º
PQ = 6.05
36º

14. Example – Decide which Law to use
Find AB in both triangles.
AB = 8.95 C C
36º
36º 14 12 12
52º A B A B