1. In this lesson, you will learn to solve any triangle, not just right triangles!
Grrrreat!
To do so, you will need to calculate trig
ratios for angle measures up to 180°.
You can use a calculator to find these values.

2. Use your calculator to find each trigonometric ratio
Use your calculator to find each trigonometric ratio. Round to the nearest hundredth.
A. tan 103°
B. cos 165°
C. sin 93°
tan 103°  –4.33
cos 165°  –0.97
sin 93°  1.00

3. The Law of Sines

4. You can use the altitude of a triangle to find a relationship between the triangle’s side lengths.
In ∆ABC, let h represent the length of the altitude from C to
From the diagram, , and
By solving for h, you find that h = b sin A and h = a sin B. So b sin A = a sin B, and
You can use another altitude to show that these ratios equal

5. You can use the Law of Sines to solve a triangle if you are given
• two angle measures and any side length
(ASA or AAS) or
• two side lengths and a non-included angle measure (SSA).

6. Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.FG
Law of Sines
Substitute the given values.
FG sin 39° = 40 sin 32°
Cross Products Property
Divide both sides by sin 39.

7. Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
mQ
Law of Sines
Substitute the given values.
Multiply both sides by 6.
Use the inverse sine function to find mQ.

8. Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.NP
Law of Sines
Substitute the given values.
NP sin 39° = 22 sin 88°
Cross Products Property
Divide both sides by sin 39°.

9. Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
mL
Law of Sines
Substitute the given values.
Cross Products Property
10 sin L = 6 sin 125°
Use the inverse sine function to find mL.

10. The Law of Cosines

11. The Law of Sines cannot be used to solve every triangle.
If you know two side lengths and the included angle measure or if you know all three side lengths, you cannot use the Law of Sines. Because you don’t have any opposite pairs of information.
Instead, you can apply the Law of Cosines.

13. Use the Law of Cosines to solve a triangle only if everything else (Pythagorean Theorem, SOH CAH TOA, or Law of Sines) doesn’t work.

14. Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.XZ
XZ2 = XY2 + YZ2 – 2(XY)(YZ)cos Y
Law of Cosines
Substitute the given values.
= – 2(35)(30)cos 110°
XZ2 
Simplify.
Find the square root of both sides.
XZ  53.3

15. Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
mT
RS2 = RT2 + ST2 – 2(RT)(ST)cos T
Law of Cosines
Substitute the given values.
72 = – 2(13)(11)cos T
49 = 290 – 286 cosT
Simplify.
Subtract 290 both sides.
–241 = –286 cosT

16. Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
mT
–241 = –286 cosT
Solve for cosT.
Use the inverse cosine function to find mT.

17. Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
mK
JL2 = LK2 + KJ2 – 2(LK)(KJ)cos K
Law of Cosines
Substitute the given values.
82 = – 2(15)(10)cos K
64 = 325 – 300 cosK
Simplify.
Subtract 325 both sides.
–261 = –300 cosK

18. Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
mK
–261 = –300 cosK
Solve for cosK.
Use the inverse cosine function to find mK.

19. What if…? Another engineer suggested using a cable attached from the top of the tower to a point 31 m from the base. How long would this cable be, and what angle would it make with the ground? Round the length to the nearest tenth and the angle measure to the nearest degree.
31 m

20. Check It Out! Example 4 Continued
Step 1 Find the length of the cable.
AC2 = AB2 + BC2 – 2(AB)(BC)cos B
Law of Cosines
Substitute the given values.
= – 2(31)(56)cos 100°
Simplify.
AC2 
Find the square root of both sides.
AC 68.6 m

21. Check It Out! Example 4 Continued
Step 2 Find the measure of the angle the cable would make with the ground.
Law of Sines
Substitute the given values.
Multiply both sides by 56.
Use the inverse sine function to find mA.