Aim: How do we evaluate and solve problems involving Law of Sines and law of Cosines?

Objective: Use the Law of Sines and the Law of Cosines to solve triangles.

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

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).

Example finding a side length 2A: Using the Law of Sines

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 sin32° Cross Products Property

Holt Geometry

Divide both sides by sin 39

Example finding an angle measure 2B: Using the Law of Sines

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. Or you could also cross multiply

Use the inverse sine function to find mQ.

Do these (2) problems below Remember read the problem, do the problem, Read the problem again and put a box or circle around final answer.

1. Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree. mX

2. Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.

AC

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. Instead, you can apply the Law of Cosines.

You can use the Law of Cosines to solve a triangle if you are given

Holt Geometry

• two side lengths and the included angle measure (SAS) or

• three side lengths (SSS).

The angle referenced in the Law of Cosines is across the equal sign from its corresponding side.

The angle and the first side in the equation must matchup

Example 3A: Using the Law of Cosines

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 is the same as y2 = z2 + x2 – 2(z)(y)cosY Law of Cosines

= 352 + 302 – 2(35)(30)cos 110° Substitute the given values y2  2843.2423 Simplify y  53.3 or XZ  53.3 Find the square root of both sides.

Example 3B: Using the Law of Cosines

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 or t2 = s2 + r2 – 2(s)(r)cos T Law of Cosines

72 = 132 + 112 – 2(13)(11)cos T Substitute the given values.

–241 = –286 cosT Simplify.

49 = 290 – 286 cosT Subtract 290 both sides.

–241 = –286 cosT

Solve for cosT.

Holt Geometry

Use the inverse cosine function to find mT.

Do these (2) problems below Remember read the problem, do the problem, Read the problem again and put a box or circle around final answer.

3. Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.

YZ

4. Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.

mR

Holt Geometry