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.



Share this page: What’s this?

Recent Articles

  1. Fundamental Geometrical Concepts | Point | Line | Properties of Lines

    Apr 18, 24 02:58 AM

    Point P
    The fundamental geometrical concepts depend on three basic concepts — point, line and plane. The terms cannot be precisely defined. However, the meanings of these terms are explained through examples.

    Read More

  2. What is a Polygon? | Simple Closed Curve | Triangle | Quadrilateral

    Apr 18, 24 02:15 AM

    What is a polygon? A simple closed curve made of three or more line-segments is called a polygon. A polygon has at least three line-segments.

    Read More

  3. Simple Closed Curves | Types of Closed Curves | Collection of Curves

    Apr 18, 24 01:36 AM

    Closed Curves Examples
    In simple closed curves the shapes are closed by line-segments or by a curved line. Triangle, quadrilateral, circle, etc., are examples of closed curves.

    Read More

  4. Tangrams Math | Traditional Chinese Geometrical Puzzle | Triangles

    Apr 18, 24 12:31 AM

    Tangrams
    Tangram is a traditional Chinese geometrical puzzle with 7 pieces (1 parallelogram, 1 square and 5 triangles) that can be arranged to match any particular design. In the given figure, it consists of o…

    Read More

  5. Time Duration |How to Calculate the Time Duration (in Hours & Minutes)

    Apr 17, 24 01:32 PM

    Duration of Time
    We will learn how to calculate the time duration in minutes and in hours. Time Duration (in minutes) Ron and Clara play badminton every evening. Yesterday, their game started at 5 : 15 p.m.

    Read More