Holt Algebra 2

5-4,5-5 The Law of Sines. Inverse Sine

Area =

Area =

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.

Write the area formula.

Substitute c sin A for h.

Holt Algebra 2

5-4,5-5 The Law of Sines. Inverse Sine

Holt Algebra 2

5-4,5-5 The Law of Sines. Inverse Sine

Example 1: Determining the Area of a Triangle

Find the area of the triangle. Round to the nearest tenth.

Area = ab sin C

≈ 4.820907073

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.

Holt Algebra 2

5-4,5-5 The Law of Sines. Inverse Sine

Holt Algebra 2

5-4,5-5 The Law of Sines. Inverse Sine

Example 2A: Using the Law of Sines

Solve the triangle. Round to the nearest tenth.

Step 1. Find the third angle measure.

mD + mE + mF = 180°

33° + mE + 28° = 180°

mE = 119°

Triangle Sum Theorem.

Substitute 33° for mD and 28° for mF.

Solve for mE.

Holt Algebra 2

5-4,5-5 The Law of Sines. Inverse Sine

Example 2A Continued

Step 2 Find the unknown side lengths.

sin D sin Fd f

=sin E sin F

e f=

sin 33° sin 28°d 15=

sin 119° sin 28°e 15=

d sin 28° = 15 sin 33° e sin 28° = 15 sin 119°

d = 15 sin 33°sin 28°

d ≈ 17.4

e = 15 sin 119°sin 28°

e ≈ 27.9Solve for the

unknown side.

Law of Sines.

Substitute.

Crossmultiply.

Holt Algebra 2

5-4,5-5 The Law of Sines. Inverse Sine

Example 2B: Using the Law of Sines

Solve the triangle. Round to the nearest tenth.

Step 2 Find the unknown side lengths.

sin P sin Qp q= sin P sin R

p r=Law of Sines.

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°sin 105°

≈ 6.5

Q

r

Holt Algebra 2

5-4,5-5 The Law of Sines. Inverse Sine

Check It Out! Example 2a

Find the missing measures in the triangle. Round to the nearest tenth.

Step 1 Find the third angle measure.

mK = 31° Solve for mK.

mH + mJ + mK = 180°

42° + 107° + mK = 180°Substitute 42° for mH

and 107° for mJ.

Holt Algebra 2

5-4,5-5 The Law of Sines. Inverse Sine

Check It Out! Example 2a Continued

Step 2 Find the unknown side lengths.

sin H sin Jh j

=sin K sin H

k h=

sin 42° sin 107°h 12=

sin 31° sin 42°k 8.4=

h sin 107° = 12 sin 42° 8.4 sin 31° = k sin 42°

h = 12 sin 42°sin 107°

h ≈ 8.4

k = 8.4 sin 31°sin 42°

k ≈ 6.5Solve for the

unknown side.

Law of Sines.

Substitute.

Crossmultiply.

Holt Algebra 2

5-4,5-5 The Law of Sines. Inverse Sine

Example 3

Law of Sines

Substitute.

Solve for sin B.

Given the measurements: a = 50, b = 20, and mA = 28°, find the other measures in the triangle. Round to the nearest tenth.

Holt Algebra 2

5-4,5-5 The Law of Sines. Inverse Sine

Example 3 Continued

Step 3 Find the other unknown measures of the triangle.

Solve for mC.

28° + 10.8° + mC = 180°

mC = 141.2°

m B = Sin-1

So: Since

Holt Algebra 2

5-4,5-5 The Law of Sines. Inverse Sine

Example 3 Continued

Now, Solve for c.

c ≈ 66.8

Law of Sines

Substitute.

Solve for c.

Holt Algebra 2

5-4,5-5 The Law of Sines. Inverse Sine

Inverse SineSymbols/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

Holt Algebra 2

5-4,5-5 The Law of Sines. Inverse Sine

Inverse Sine

22

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?

Holt Algebra 2

5-4,5-5 The Law of Sines. Inverse Sine

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

Holt Algebra 2

5-4,5-5 The Law of Sines. Inverse Sine

ExamplesEvaluate the following:

a) b)

c)

)1sin(Arcradiansin 2

3sin 1

1000sinsin 1