Subscribe to our YouTube channel for the latest videos, updates, and tips.


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. 8 Times Table | Multiplication Table of 8 | Read Eight Times Table

    May 18, 25 04:33 PM

    Printable eight times table
    In 8 times table we will memorize the multiplication table. Printable multiplication table is also available for the homeschoolers. 8 × 0 = 0 8 × 1 = 8 8 × 2 = 16 8 × 3 = 24 8 × 4 = 32 8 × 5 = 40

    Read More

  2. Worksheet on Average | Word Problem on Average | Questions on Average

    May 17, 25 05:37 PM

    In worksheet on average interest we will solve 10 different types of question. Find the average of first 10 prime numbers. The average height of a family of five is 150 cm. If the heights of 4 family

    Read More

  3. How to Find the Average in Math? | What Does Average Mean? |Definition

    May 17, 25 04:04 PM

    Average 2
    Average means a number which is between the largest and the smallest number. Average can be calculated only for similar quantities and not for dissimilar quantities.

    Read More

  4. Problems Based on Average | Word Problems |Calculating Arithmetic Mean

    May 17, 25 03:47 PM

    Here we will learn to solve the three important types of word problems based on average. The questions are mainly based on average or mean, weighted average and average speed.

    Read More

  5. Rounding Decimals | How to Round a Decimal? | Rounding off Decimal

    May 16, 25 11:13 AM

    Round off to Nearest One
    Rounding decimals are frequently used in our daily life mainly for calculating the cost of the items. In mathematics rounding off decimal is a technique used to estimate or to find the approximate

    Read More