cos 3A in Terms of A

We will learn how to express the multiple angle of cos 3A in terms of A or cos 3A in terms of cos A.

Trigonometric function of cos 3A in terms of cos A is also known as one of the double angle formula.

If A is a number or angle then we have, cos 3A = 4 cos^3 A - 3 cos A

Now we will proof the above multiple angle formula step-by-step.

Proof: cos 3A

= cos (2A + A)

= cos 2A cos A - sin 2A sin A

= (2 cos^2 A - 1) cos A - 2 sin A cos A ∙ sin A

= 2 cos^3 A - cos A - 2 cos A (1 - cos^2 A)

= 2 cos^3 A - cos A - 2 cos A + 2 cos^3 A

= 4 cos^3 A - 3 cos A

Therefore,  cos 3A = 4 cos^3 A - 3 cos A             Proved

Note:  (i) In the above formula we should note that the angle on the R.H.S. of the formula is one-third of the angle on L.H.S. Therefore, cos 120° = 4 cos^3 40° - 3 cos 40°.

(ii) To find the formula of cos 3A in terms of A or cos 3A in terms of cos A we have use cos 2A = 2cos^2 A - 1.


Now, we will apply the formula of multiple angle of cos 3A in terms of A or cos 3A in terms of cos A to solve the below problems.

1. Prove that: cos 6A = 32 cos^6 A - 48 cos^4 A + 18 cos^2 A - 1

Solution:

L.H.S. = cos 6A

         = 2 cos^2 3A - 1, [Since we know that, cos 2θ = 2 cos^2 θ - 1]

         = 2(4 cos^3 A - 3 cos A)^2 - 1

         = 2 (16 cos^ 6 A + 9 cos^2 A - 24 cos^2 A) - 1

         = 32 cos^6 A – 48 cos^4 A + 18 cos^2 A - 1 = R.H.S.

 

2. Show that, 32 sin^6 θ = 10 - 15 cos 2θ + 6 cos 4θ - cos 6θ

Solution:               

L.H.S = 32 sin^6 θ

         = 4 ∙ (2 sin^2 θ)^3

         = 4 (1 - cos 2θ)^3

         = 4 [1 - 3 cos 2θ + 3 ∙ cos^2 2θ - cos^3 2θ]

         = 4 - 12 cos^2 θ + 12 cos^2 2θ - 4 cos^3 2θ

         = 4 - 12 cos 2θ + 6 ∙ 2 cos^2 2θ   - [cos 3 ∙ (2θ) + 3 cos 2θ]

         [Since, cos 3A = 4 cos^3 A - 3 cos A

         Therefore, 4 cos^3 A = cos 3A + 3 cos A]

⇒ 4 cos^3 2θ = cos 3 ∙ (2θ) + 3 cos 2θ, (replacing A by 2θ)

                   = 4 - 12 cos 2θ + 6 (1 + cos 4θ) - cos 6θ - 3 cos 2θ

                   = 10 - 15 cos 2θ + 6 cos 4θ - cos 6θ = R.H.S.                 Proved

 

3. Prove that: cos A cos (60 - A) cos (60 + A) = ¼ cos 3A

Solution:

L.H.S. = cos A ∙ cos (60 - A) cos (60 + A)

         = cos A ∙ (cos^2 60 - sin^2 A), [Since we know that cos (A + B) cos (A - B)          = cos ^2 A - sin ^2 B]

         = cos A (¼ - sin^2 A)

         = cos A (¼ - (1 - cos^2 A))

         = cos A (-3/4 + cos ^2 A)

         = ¼ cos A (-3 + 4 cos^2 A)

         = ¼(4 cos^3A - 3 cos A)

         = ¼ cos 3A = R.H.S.                        Proved

 Multiple Angles






11 and 12 Grade Math

From cos 3A in Terms of A to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

  1. Expanded form of Decimal Fractions |How to Write a Decimal in Expanded

    Jul 22, 24 03:27 PM

    Expanded form of Decimal
    Decimal numbers can be expressed in expanded form using the place-value chart. In expanded form of decimal fractions we will learn how to read and write the decimal numbers. Note: When a decimal is mi…

    Read More

  2. Worksheet on Decimal Numbers | Decimals Number Concepts | Answers

    Jul 22, 24 02:41 PM

    Worksheet on Decimal Numbers
    Practice different types of math questions given in the worksheet on decimal numbers, these math problems will help the students to review decimals number concepts.

    Read More

  3. Decimal Place Value Chart |Tenths Place |Hundredths Place |Thousandths

    Jul 21, 24 02:14 PM

    Decimal place value chart
    Decimal place value chart are discussed here: The first place after the decimal is got by dividing the number by 10; it is called the tenths place.

    Read More

  4. Thousandths Place in Decimals | Decimal Place Value | Decimal Numbers

    Jul 20, 24 03:45 PM

    Thousandths Place in Decimals
    When we write a decimal number with three places, we are representing the thousandths place. Each part in the given figure represents one-thousandth of the whole. It is written as 1/1000. In the decim…

    Read More

  5. Hundredths Place in Decimals | Decimal Place Value | Decimal Number

    Jul 20, 24 02:30 PM

    Hundredths Place in Decimals
    When we write a decimal number with two places, we are representing the hundredths place. Let us take plane sheet which represents one whole. Now, we divide the sheet into 100 equal parts. Each part r…

    Read More