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. Formation of Square and Rectangle | Construction of Square & Rectangle

    Jul 15, 25 02:46 AM

    Construction of a Square
    In formation of square and rectangle we will learn how to construct square and rectangle. Construction of a Square: We follow the method given below. Step I: We draw a line segment AB of the required…

    Read More

  2. 5th Grade Quadrilaterals | Square | Rectangle | Parallelogram |Rhombus

    Jul 15, 25 02:01 AM

    Square
    Quadrilaterals are known as four sided polygon.What is a quadrilateral? A closed figure made of our line segments is called a quadrilateral. For example:

    Read More

  3. Formation of Numbers | Smallest and Greatest Number| Number Formation

    Jul 14, 25 01:53 AM

    In formation of numbers we will learn the numbers having different numbers of digits. We know that: (i) Greatest number of one digit = 9,

    Read More

  4. 5th Grade Geometry Practice Test | Angle | Triangle | Circle |Free Ans

    Jul 14, 25 01:53 AM

    Name the Angles
    In 5th grade geometry practice test you will get different types of practice questions on lines, types of angle, triangles, properties of triangles, classification of triangles, construction of triang…

    Read More

  5. 5th Grade Circle Worksheet | Free Worksheet with Answer |Practice Math

    Jul 11, 25 02:14 PM

    Radii of the circRadii, Chords, Diameters, Semi-circles
    In 5th Grade Circle Worksheet you will get different types of questions on parts of a circle, relation between radius and diameter, interior of a circle, exterior of a circle and construction of circl…

    Read More