1. Area =
Write the area formula.
Area =
Substitute c sin A for h.
This formula allows you to determine the area of a triangle if you know the lengths of two of its sides and the measure of the angle between them.

3. Example 1: Determining the Area of a Triangle
Find the area of the triangle. Round to the nearest tenth.
Area = ab sin C
Write the area formula.
Substitute 3 for a, 5 for b, and 40° for C.
Use a calculator to evaluate the expression. ≈
The area of the triangle is about 4.8 m2.

5. Example 2A: Using the Law of Sines
Solve the triangle. Round to the nearest tenth.
Step 1. Find the third angle measure.
mD + mE + mF = 180°
Triangle Sum Theorem.
Substitute 33° for mD and 28° for mF.
33° + mE + 28° = 180°
mE = 119°
Solve for mE.

6. Example 2A Continued
Step 2 Find the unknown side lengths.
sin D
sin F d f =
sin E
sin F e f =
Law of Sines.
sin 33°
sin 28° d 15 =
sin 119°
sin 28° e 15 =
Substitute.
Cross
multiply.
d sin 28° = 15 sin 33°
e sin 28° = 15 sin 119°
e =
15 sin 119°
sin 28°
e ≈ 27.9
d =
15 sin 33°
sin 28°
d ≈ 17.4
Solve for the
unknown side.

7. Example 2B: Using the Law of SinesQ r
Solve the triangle. Round to the nearest tenth.
Step 2 Find the unknown side lengths.
sin P
sin Q p q =
Law of Sines.
sin P
sin R p r =
sin 105°
sin 36° 10 q =
sin 105°
sin 39° 10 r =
Substitute.
q =
10 sin 36°
sin 105°
≈ 6.1
r =
10 sin 39°
≈ 6.5

8. Check It Out! Example 2a
Find the missing measures in the triangle. Round to the nearest tenth.
Step 1 Find the third angle measure.
mH + mJ + mK = 180°
Substitute 42° for mH and 107° for mJ.
42° + 107° + mK = 180°
mK = 31°
Solve for mK.

9. Check It Out! Example 2a Continued
Step 2 Find the unknown side lengths.
sin H
sin J h j =
sin K
sin H k h =
Law of Sines.
sin 42°
sin 107° h 12 =
sin 31°
sin 42° k
8.4 =
Substitute.
Cross
multiply.
h sin 107° = 12 sin 42°
8.4 sin 31° = k sin 42°
k =
8.4 sin 31°
sin 42°
k ≈ 6.5
h =
12 sin 42°
sin 107°
h ≈ 8.4
Solve for the
unknown side.

10. Example 3
Given the measurements: a = 50, b = 20, and mA = 28°, find the other measures in the triangle. Round to the nearest tenth.
Law of Sines
Substitute.
Solve for sin B.

11. Example 3 Continued
So: Since
m B = Sin-1
Step 3 Find the other unknown measures of the triangle.
Solve for mC.
28° ° + mC = 180°
mC = 141.2°

12. Example 3 Continued
Now, Solve for c.
Law of Sines
Substitute.
Solve for c.
c ≈ 66.8

13. Inverse Sine Symbols/Notation: sin-1 or Arcsin
Always used to find an angle given the ratio of sides oppostie/hypotenuse
In order to view the inverse Sine as a function, we must review what we know about functions and their inverses

14. Inverse Sine Soooo….What interval does this?
Q: How can we determine if a graph represents a function?
Q: How can we determine if the graph of a function has an Inverse that is also a function?
So: We restrict the domain…
1)Include 1st quadrant
2)Include the entire range of the function
3)Continuous interval
Soooo….What interval does this?

15. Domain and Range Restricted Sine Function: Domain: -π/2 ≤ x ≤ π/2
Range: -1 ≤ y ≤ 1
Inverse Sine Function:
Domain: -1 ≤ x ≤ 1
Range: -π/2 ≤ x ≤ π/2

16. Examples
Evaluate the following:
a) b) c)