Multiply by 13

==>

Rewrite to place the radical on one side before squaring
the two sides of the equation.

Check using Mathcad

4. Find the domain
and range of

Solution:

Domain:

==>

Range:

Use the fact that

(see Figure 4)

==>

==>

and

(see Figure 5)

Substitute in (1)

Figure 4

Figure 5

We now equation the expression above to 1 and solve
for x

==>

a)

False

Let x = 1

then

and

undefined

b)

False

The secant of a negative number less than minus one
is negative.

c)

True

Because 3 is in the domain of cos and in the range of
cos-1.

d)

False

Because 3 is not in the domain of cos-1.

e)

False

Because 4p
/ 3 is in the domain of tan but not in the range of tan-1
.

recall:

Remark:

The general rule for deciding on the possible values
of x in a composite function of the form:

outside_function(inside_function(x))

is: x must be in the domain
of inside_function and in the
range of outside_function.

Multiply the inequality by 2.

Subtract p
from all sides.

5. Solve the equation
, where

Solution:

Thus,

==>

or

==>

or

6. Answer with
true or false

==>

==>

(see Figure 2)

Please ignore

the guys above.

They are used

to draw the

graphs.

Note:

The coordinates of the graphs are not scaled equally.

13

5

12

3

4

5

Figure 1

Figure 2

Thus,

Math 002-041

Solutions
to Cal Problems
6.5 - 6.6

A. Farhat

1. Find the exact
value of
.

Solution:

Let

and

This substitution transforms the question to finding
the value of
.

5

==>

==>

(see Figure 1)

Thus,

3. Solve the equation
.

Solution:

Take sine of both sides and evaluate
.

Let

and

Thus, we need to find

(1)

==>

==>

Check using Mathcad:

This says that Mathcad is unable to find the exact value.

This says that Mathcad can find the approximate value.

2. Verify the
identity
.

Solution:

LHS = Left Hand Side

Let

This substitution transforms the problem to that of
finding

sin(2 a)
= 2 sin(a)
cos(a).

==>

==> (See Figure 3)

and

Figure 3