Identities Involving Sines and Cosines

Identities involving sines and cosines of multiples or submultiples of the angles involved.

To prove the identities involving sines and cosines we use the following algorithm.

Step I: Convert the sum of first two terms as product by using one of the following formulae:

sin C + sin D = 2 sin \(\frac{C + D}{2}\) cos \(\frac{C - D}{2}\)

sin C - sin D = 2 cos \(\frac{C + D}{2}\) sin \(\frac{C - D}{2}\)

cos C + cos D = 2 cos \(\frac{C + D}{2}\) cos \(\frac{C - D}{2}\)

cos C - cos D  = - 2 sin \(\frac{C + D}{2}\) sin \(\frac{C - D}{2}\)

Step II: In the product obtain in step II replace the sum of two angles in terms of the third by using the given relation.

Step III: Expand the third term by using one of the following formulas:

sin 2θ = 2 sin θ cos θ,

cos 2θ = 2 cos\(^{2}\) θ - 1

cos 2θ = 1 - 2 sin\(^{2}\) θ etc.


Step IV: Take the common factor outside.

Step V: Express the trigonometric ratio of the single angle in terms of the remaining angles.

Step VI: Use one of the formulas given in step I to convert the sum into product.


Examples on identities involving sines and cosines:

1. If A + B + C = π prove that, sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C.

Solution:

L.H.S. = (sin 2A + sin 2B) + sin 2C

= 2 sin \(\frac{2A + 2B}{2}\) cos \(\frac{2A - 2B}{2}\)+ sin 2C

= 2 sin (A + B) cos (A - B) + sin 2C

= 2 sin (π - C) cos (A - B) + sin 2C, [Since, A + B + C = π ⇒ A + B =  π - C]

= 2 sin C cos (A - B) + 2 sin C cos C, [Since  sin (π - C) =  sin C]

= 2 sin C [cos (A - B) + cos C], taking common 2 sin C

= 2 sin C [cos (A - B) + cos {π - (A + B)}], [Since A + B + C = π ⇒ C = π - (A + B)]

= 2 sin C [cos (A - B) - cos (A + B)], [Since cos {π - (A + B)} = - cos (A + B)]

= 2 sin C [2 sin A sin B], [Since cos (A - B) - cos (A + B) = 2 sin A sin B]

= 4 sin A sin B sin C.        Proved.



2. If A + B + C = π prove that, cos 2A + cos 2B - cos 2C = 1- 4 sin A sin B cos C.

Solution:

L.H.S. = cos 2A + cos 2B - cos 2C

= (cos 2A + cos 2B) - cos 2C

= 2 cos \(\frac{2A + 2B}{2}\) cos \(\frac{2A - 2B}{2}\) - cos 2C

= 2 cos (A + B) cos (A- B) - cos 2C

= 2 cos (π - C) cos (A- B) - cos 2C, [Since we know A + B + C = π ⇒A + B = π – C]

= - 2 cos C cos (A - B) – (2 cos\(^{2}\) C - 1), [Since cos (π - C) = - cos C]

= - 2 cos C cos (A - B) - 2 cos\(^{2}\) C + 1

= - 2 cos C [cos (A - B) + cos C] + 1

= -2 cos C [cos (A - B) - cos (A + B)] + 1, [Since cos C = - cos (A + B)]

= -2 cos C [2 sin A sin B] + 1, [Since cos (A - B) - cos (A + B) = 2 sin A sin B]

= 1 - 4 sin A sin B cos C.        Proved.

 Conditional Trigonometric Identities






11 and 12 Grade Math

From Identities Involving Sines and Cosines 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