Trig Ratios Proving Problems

In trig ratios proving problems we will learn how to proof the questions step-by-step using trigonometric identities.

1. If (1 + cos A)( 1 + cos B)( 1 + cos C) = (1 - cos A)( 1 - cos B)( 1 - cos C) then prove that each side = ± sin A sin B sin C.

Solution:  Let, (1 + cos A) (1 + cos B) (1 + cos C) = k         …. (i)

Therefore, according to the problem,

(1 - cos A) (1 - cos B) (1 - cos C) = k                         ….. (ii)

Now multiplying both sides of (i) and (ii) we get,

(1 + cos A)(1 + cos B)(1 + cos C)(1 - cos A)(1 - cos B)(1 - cos C) = k2

⇒ k2 = (1 - cos2 A) (1 - cos2 B) (1 - cos2 C)

⇒ k2 = sin2 A sin2 B sin2 C

 k = ± sin A sin B sin C.

Therefore, each side of the given condition

= k = ± sin A sin B  sin C 
                                           Proved.


More solved examples on trig ratios proving problems.

2. If un = cosn θ + sinn θ then prove that, 2u6 - 3u4 + 1 = 0.

Solution:

Since, un = cosn θ + sinn θ

Therefore, u6 = cos6 θ + sin6 θ

⇒ u6 = (cos2 θ)3 + (sin2 θ)3

⇒ u6 = (cos2 θ + sin2 θ)3 - 3 cos2 θ ∙ sin2 θ (cos2 θ + sin2 θ)

⇒ u6 = 1 - 3cos2 θ sin2 θ and u4 = cos4 θ + sin4 θ

⇒ u4 = (cos2 θ)2 + (sin2 θ)2

⇒ u4 = (cos2 θ + sin2 θ)2 - 2 cos2 θ sin2 θ

⇒ u4 = 1 - 2 cos2 θ sin2 θ

Therefore,

2u6 - 3u4 + 1

= 2(1 - 3cos2 θ sin2 θ) - 3(1 - 2 cos2 θ sin2 θ) + 1

= 2 - 6 cos2 θ sin2 θ - 3 + 6 cos2 θ sin2 θ + 1

= 0.

Therefore, 2u6 - 3u4 + 1 = 0.

                                           Proved.


3. If a sin θ - b cos θ = c then prove that, a cos θ + b sin θ = ± √(a2 + b2 - c2).

Solution:

Given: a sin θ - b cos θ = c

⇒ (a sin θ - b cos θ)2 = c2, [Squaring both sides]

⇒ a2 sin2 θ + b2 cos2 θ - 2ab sin θ cos θ = c2

⇒ - a2 sin2 θ - b2 cos2 θ + 2ab sin θ cos θ = - c2

⇒ a2 - a2 sin2 θ + b2 - b2 cos2 θ + 2ab sin θ cos θ = a2 + b2 - c2

⇒ a2(1 - sin2 θ) + b2(1 - cos2 θ) + 2ab sin θ cos θ = a2 + b2 - c2

⇒ a2 cos2 θ + b2 sin2 θ + 2 ∙ a cos θ ∙ b sin θ = a2 + b2 - c2

⇒ (a cos θ + b sin θ)2 = a2 + b2 - c2

Now taking square root on both the sides we get,

⇒ a cos θ + b sin θ = ± √(a2 + b2 - c2).

                                                      Proved.


The above three trig ratios proving problems will help us to solve more basic problems on T-ratio.

Basic Trigonometric Ratios 

Relations Between the Trigonometric Ratios

Problems on Trigonometric Ratios

Reciprocal Relations of Trigonometric Ratios

Trigonometrical Identity

Problems on Trigonometric Identities

Elimination of Trigonometric Ratios 

Eliminate Theta between the equations

Problems on Eliminate Theta 

Trig Ratio Problems

Proving Trigonometric Ratios

Trig Ratios Proving Problems

Verify Trigonometric Identities 






10th Grade Math

From Trig Ratios Proving Problems 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