# TrigonometryFinding the Range of a Trigonometric Function

#### melissax

##### New member
Hello, I have some questions and i couldnt solve them can you help me?

If y=5cos(x)+3 then what is the heap of ?

(a) All real numbers
(b) alpha<= y <= alpha
(c) -2 <= y <= 10
( d)-2 <= y <= 8 What is the solution?

Thank you.

Last edited by a moderator:

#### MarkFL

Staff member
re: Finding the Range of a Trignometric Function

To find the range of the given sinusoid, I would use this method:

$\displaystyle -1\le\cos(x)\le1$

$\displaystyle -A\le A\cos(x)\le A$

$\displaystyle B-A\le A\cos(x)+B\le B+A$

Can you apply this procedure to the function you are given?

#### melissax

##### New member
re: Finding the Range of a Trignometric Function

Thank you very much
You showed me path, i will apply.

#### melissax

##### New member
re: Finding the Range of a Trignometric Function

-1<=5*Cos(x)+3<=1
-5<=5*Cos(x)+3<=5
-5-3<=5Cos(x)<=5-3
-8<=5Cos(x)<=2

As i understand between -2 and 8 but how i can show?

#### MarkFL

Staff member
re: Finding the Range of a Trignometric Function

You have found the correct range, but what you actually want to do is this:

Begin with the fact that the cosine function varies from -1 to 1:

$\displaystyle -1\le\cos(x)\le1$

Multiply through by the given amplitude of 5:

$\displaystyle -5\le 5\cos(x)\le5$

Add through by the given vertical displacement of 3:

$\displaystyle 3-5\le 5\cos(x)+3\le3+5$

Simplify:

$\displaystyle -2\le 5\cos(x)+3\le8$

And this demonstrates the range is [-2,8].

#### melissax

##### New member
re: Finding the Range of a Trignometric Function

I am sory. You showed me path but i wrote wrong.
When i solved second then i saw?

Thank you. You are great teacher.

#### MarkFL

Glad to help out, and welcome to the forum! 