Notes 5-4

Section 5-4 T h e L a w o f S i n e s

Theorem: SAS Area Formula for a triangle

Theorem: SAS Area Formula for a triangle The area for any triangle is half the product of the lengths of two sides and the sine of the angle between them

Theorem: SAS Area Formula for a triangle The area for any triangle is half the product of the lengths of two sides and the sine of the angle between them This leads us to our next theorem...

Theorem: Law of Sines

Theorem: Law of Sines In any triangle ABC: sin A sin B sinC = = a b c

Theorem: Law of Sines In any triangle ABC: sin A sin B sinC = = a b c C b a A B c

Example 1 From a speciﬁc point in a school classroom, students can see the the top of a cell phone tower if they look 41° west of north. Traveling 4.1 miles due west from the school, the same cell phone tower is seen in the direction 26° west of north. How far is the school from the top of the cell phone tower?

Example 1 From a speciﬁc point in a school classroom, students can see the the top of a cell phone tower if they look 41° west of north. Traveling 4.1 miles due west from the school, the same cell phone tower is seen in the direction 26° west of north. How far is the school from the top of the cell phone tower? First, we need to set up our triangle.

Example 1 From a speciﬁc point in a school classroom, students can see the the top of a cell phone tower if they look 41° west of north. Traveling 4.1 miles due west from the school, the same cell phone tower is seen in the direction 26° west of north. How far is the school from the top of the cell phone tower? First, we need to set up our triangle. Tower 15° x 116° 49° School 4.1 miles

Example 1 From a speciﬁc point in a school classroom, students can see the the top of a cell phone tower if they look 41° west of north. Traveling 4.1 miles due west from the school, the same cell phone tower is seen in the direction 26° west of north. How far is the school from the top of the cell phone tower? First, we need to set up our triangle. Tower sin116° sin15° = 15° x x 4.1 116° 49° School 4.1 miles

Example 1 From a speciﬁc point in a school classroom, students can see the the top of a cell phone tower if they look 41° west of north. Traveling 4.1 miles due west from the school, the same cell phone tower is seen in the direction 26° west of north. How far is the school from the top of the cell phone tower? First, we need to set up our triangle. Tower sin116° sin15° = 15° x x 4.1 4.1sin116° 116° 4.1 miles 49° School x= sin15°

Example 1 From a speciﬁc point in a school classroom, students can see the the top of a cell phone tower if they look 41° west of north. Traveling 4.1 miles due west from the school, the same cell phone tower is seen in the direction 26° west of north. How far is the school from the top of the cell phone tower? First, we need to set up our triangle. Tower sin116° sin15° = 15° x x 4.1 4.1sin116° 116° 4.1 miles 49° School x= sin15° x ≈ 14.23796149

Example 1 From a speciﬁc point in a school classroom, students can see the the top of a cell phone tower if they look 41° west of north. Traveling 4.1 miles due west from the school, the same cell phone tower is seen in the direction 26° west of north. How far is the school from the top of the cell phone tower? First, we need to set up our triangle. Tower sin116° sin15° = 15° x x 4.1 4.1sin116° 116° 4.1 miles 49° School x= sin15° x ≈ 14.23796149 miles

Example 2 In  XYZ , XY = 12, YZ = 10, and m∠X = 38°. Find m∠Z.

Example 2 In  XYZ , XY = 12, YZ = 10, and m∠X = 38°. Find m∠Z. X 38° 12 Y Z 10

Example 2 In  XYZ , XY = 12, YZ = 10, and m∠X = 38°. Find m∠Z. sin 38° sin Z X = 10 12 38° 12 Y Z 10

Example 2 In  XYZ , XY = 12, YZ = 10, and m∠X = 38°. Find m∠Z. sin 38° sin Z X = 10 12 38° 12 12sin 38° = sin Z 10 Y Z 10

Example 2 In  XYZ , XY = 12, YZ = 10, and m∠X = 38°. Find m∠Z. sin 38° sin Z X = 10 12 38° 12 12sin 38° = sin Z 10 Y Z 10 sin Z ≈ .7387937704

Example 2 In  XYZ , XY = 12, YZ = 10, and m∠X = 38°. Find m∠Z. sin 38° sin Z X = 10 12 38° 12 12sin 38° = sin Z 10 Y Z 10 sin Z ≈ .7387937704 ⎛ 12sin 38° ⎞ sin −1 ( ) −1 sin Z ≈ sin ⎜ ⎝ 10 ⎟ ⎠

Example 2 In  XYZ , XY = 12, YZ = 10, and m∠X = 38°. Find m∠Z. sin 38° sin Z X = 10 12 38° 12 12sin 38° = sin Z 10 Y Z 10 sin Z ≈ .7387937704 ⎛ 12sin 38° ⎞ sin −1 ( ) −1 sin Z ≈ sin ⎜ ⎝ 10 ⎟ ⎠ Z ≈ 47.62876444°

Example 2 In  XYZ , XY = 12, YZ = 10, and m∠X = 38°. Find m∠Z. sin 38° sin Z X = 10 12 38° 12 12sin 38° = sin Z 10 Y Z 10 sin Z ≈ .7387937704 ⎛ 12sin 38° ⎞ sin −1 ( ) sin Z ≈ sin ⎜ ⎝ −1 10 ⎟ ⎠ Z ≈ 47.62876444° or

Example 2 In  XYZ , XY = 12, YZ = 10, and m∠X = 38°. Find m∠Z. sin 38° sin Z X = 10 12 38° 12 12sin 38° = sin Z 10 Y Z 10 sin Z ≈ .7387937704 ⎛ 12sin 38° ⎞ sin −1 ( ) sin Z ≈ sin ⎜ ⎝ −1 10 ⎟ ⎠ Z ≈ 47.62876444° or Z ≈ 132.3712356°

Example 3 In  XYZ , XY = 12, YZ = 20, and m∠X = 38°. Find m∠Z. X sin 38° sin Z = 38° 20 12 12 Y Z 20

Example 3 In  XYZ , XY = 12, YZ = 20, and m∠X = 38°. Find m∠Z. X sin 38° sin Z = 38° 20 12 12 12sin 38° = sin Z Y Z 20 20

Example 3 In  XYZ , XY = 12, YZ = 20, and m∠X = 38°. Find m∠Z. X sin 38° sin Z = 38° 20 12 12 12sin 38° = sin Z Y Z 20 20 −1 ⎛ 12sin 38° ⎞ sin −1 ( ) sin Z = sin ⎜ ⎝ 20 ⎠ ⎟

Example 3 In  XYZ , XY = 12, YZ = 20, and m∠X = 38°. Find m∠Z. X sin 38° sin Z = 38° 20 12 12 12sin 38° = sin Z Y Z 20 20 −1 ⎛ 12sin 38° ⎞ sin −1 ( ) sin Z = sin ⎜ ⎝ 20 ⎠ ⎟ Z ≈ 21.67842645°

Example 3 In  XYZ , XY = 12, YZ = 20, and m∠X = 38°. Find m∠Z. X sin 38° sin Z = 38° 20 12 12 12sin 38° = sin Z Y Z 20 20 −1 ⎛ 12sin 38° ⎞ sin −1 ( ) sin Z = sin ⎜ ⎝ 20 ⎠ ⎟ Z ≈ 21.67842645° or

Example 3 In  XYZ , XY = 12, YZ = 20, and m∠X = 38°. Find m∠Z. X sin 38° sin Z = 38° 20 12 12 12sin 38° = sin Z Y Z 20 20 −1 ⎛ 12sin 38° ⎞ sin −1 ( ) sin Z = sin ⎜ ⎝ 20 ⎠ ⎟ Z ≈ 21.67842645° or Z ≈ 152.3215736°

Example 4 In  XYZ , XY = 12, YZ = 5, and m∠X = 38°. Find m∠Z. X 38° Wait a minute! I can’t get an answer?! 12 Y Z 5

Example 4 In  XYZ , XY = 12, YZ = 5, and m∠X = 38°. Find m∠Z. X 38° Wait a minute! I can’t get an answer?! 12 Why not? Y Z 5

Example 4 In  XYZ , XY = 12, YZ = 5, and m∠X = 38°. Find m∠Z. X 38° Wait a minute! I can’t get an answer?! 12 Why not? Y Z 5 12sin 38° ≈ 5

Example 4 In  XYZ , XY = 12, YZ = 5, and m∠X = 38°. Find m∠Z. X 38° Wait a minute! I can’t get an answer?! 12 Why not? Y Z 5 12sin 38° ≈ 1.477587541 5

Example 4 In  XYZ , XY = 12, YZ = 5, and m∠X = 38°. Find m∠Z. X 38° Wait a minute! I can’t get an answer?! 12 Why not? Y Z 5 12sin 38° ≈ 1.477587541 That’s outside the domain. 5

If we have two angles, how many possibilities will we look for in the answers?

If we have two angles, how many possibilities will we look for in the answers? Only one.

If we have two angles, how many possibilities will we look for in the answers? Only one. If we have two sides, how many possibilities will we look for in the answers?

If we have two angles, how many possibilities will we look for in the answers? Only one. If we have two sides, how many possibilities will we look for in the answers? There are two possible answers.

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Notes 5-4

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