Exercise 5 Page 612

We will check if the following statement is always true, sometimes true, or never true. 3x-5=2(x+1)+xTo do this, we will simplify the right-hand side using the Distributive Property.

Notice that, for any value of x, right-hand side is always greater than the left-hand side of this equation. This means that the sides are never equal.

Value of x	3x-5	3x+2	3x-5? =3x+2
1	3( 1)-5=-2	3( 1)+2=5	- 2≠ 5 *
50	3( 50)-5=145	3( 50)+2=152	145≠ 152 *
- 100	3( -100)-5=-305	3( -100)+2=-298	-305≠ -298 *

We found that the given statement is never true.

We are asked to check whether the given statement is always true, sometimes true, or never true. sin (x)=cos(π/2-x) Let's evaluate each side for a few special angles.

Angle x	sin (x)	cos(π/2-x)	sin (x)= cos(π/2-x)
π/6	sin π/6=1/2	cos(π/2- π/6)=cos π/3=1/2	1/2=1/2 ✓
π/4	sin π/4=sqrt(2)/2	cos(π/2- π/4)=cos π/2=sqrt(2)/2	sqrt(2)/2=sqrt(2)/2 ✓
π/3	sin π/3=sqrt(3)/2	cos(π/2- π/3)=cos π/6=sqrt(3)/2	sqrt(3)/2=sqrt(3)/2 ✓
π/2	sin π/2=1	cos(π/2- π/2)=cos 0=1	1=1 ✓

We can see that for the special angles, the left- and right-hand sides of the given statement are equal. This means that the statement is most likely always true.

We will determine whether the given statement is always true, sometimes true, or never true. tan x =1 Let's evaluate tan x for a few special angles.

Angle x	tan x	tan x ? = 1
0	tan 0=0	0≠ 1 *
π/6	tan π/6=sqrt(3)/3	sqrt(3)/3≠ 1 *
π/4	tan π/4=1	1=1 ✓

We can see that for some values of x the given statement is not true and for x= π4 the given statement is true. This means that the given statement is sometimes true. Now we will find all values of x for which tan x =1. Let's take a look at the graph of the tangent function.

We can see that if we add or subtract a multiple of π from π4, then the tangent of that angle is equal to 1.

Angle x	tan x
π/4-π	tan ( π/4-π)=1
π/4	tan ( π/4)=1
π/4+π	tan ( π/4+π)=1
π/4+2π	tan ( π/4+2π)=1

Let's try to write all angles for which the given statement is true using one concise expression. We will use n to represent any integer. Angles for Which the Given Statement Is True: [0.4 em] x=π/4+nπ