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# if angle a and angle b are acute angles such that cos a is equal to cos b then show that angle a is equal to angle b

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### Mohammed

Guys, does anyone know the answer?

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## If angle A and angle B are acute angles such that cosA=cosB, then show that angleA=angle B.

If angle A and angle B are acute angles such that cosA=cosB, then show that angleA=angle B.

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If angle A and angle B are acute angles such that cosA=cosB, then show that angleA=angle B.

Question

If angle A and angle B are acute angles such that cosA=cosB, then show that angleA=angle B.

Open in App Solution

Suggest Corrections 17 SIMILAR QUESTIONS

Q.

If ∠A and ∠B are acute angles such that cos A = cos B, then show that

∠A = ∠B.

Q. If ∠A and ∠P are acute angles such that tan A = tan P, then show that ∠A = ∠P.Q. If

∠ A and ∠ B

are acute angles such that

cos A = cos B then prove that ∠ A = ∠ B .

Q. Question 6

If ∠ A and ∠

B are acute angles such that cos A = cos B, then show that

∠ A = ∠ B.

Q. In triangle ABC and triangle XYZ if angle A and angle X are acute angles such that cosA=cosX then show that angle A=angle X

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## If A and B are acute angles such that cos A = cos B then prove that A = B .

Click here👆to get an answer to your question ✍️ If A and B are acute angles such that cos A = cos B then prove that A = B .

Question

## If ∠A and ∠B are acute angles such that cosA=cosB then prove that ∠A=∠B.

Medium Open in App Solution Verified by Toppr

According to the question:-

Let cosA= Hypotenuse sideadjacentA ​ AB AC ​ Similarly, cosB= Hypotenuse sideadjacentB ​ = AB BC ​ Given that cosA=cosB AB AC ​ = AB BC ​ AC=BC In triangle,

angles opposite equal side are equal

∠B=∠A

1078 57

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## If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B

If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B. Using cos A and cos B, we can find ratio of the length of two sides of the right-angled triangle with respective angles.

## If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B

Solution:

Using the basic trigonometric ratios, we can solve this problem.

In the right-angled triangle ABC as shown below, ∠A and ∠B are acute angles and ∠C is right angle.

cos A = side adjacent to ∠A / hypotenuse = AC/AB

cos B = side adjacent to ∠B / hypotenuse = BC/AB

Given that cos A = cos B

Therefore, AC/AB = BC/AB

AC = BC

Hence, ∠A = ∠B (angles opposite to equal sides of a triangle are equal.)

Let's look into an alternative approach to solve the question.

Let us consider a triangle ABC in which CO ⊥ AB.

It is given that cos A = cos B

AO/AC = BO/BC AO/BO = AC/BC

Let AO/BO = AC/BC = k

AO = k × BO ...(i) AC = k × BC ...(ii)

By applying Pythagoras theorem in ΔCAO and ΔCBO, we get

AC2 = AO2 + CO2 (from ΔCAO)

CO2 = AC2 - AO2 ...(iii)

BC2 = BO2 + CO2 (from ΔCBO)

CO2 = BC2 - BO2 ...(iv)

From equation (iii) and equation (iv), we get

AC2 - AO2 = BC2 - BO2

(kBC)2 - (kBO)2 = BC2 - BO2 [From equation (i) and (ii)]

k2BC2 - k2BO2 = BC2 - BO2

k2 (BC2 - BO2) = BC2 - BO2

k2 = (BC2 - BO2)/(BC2 - BO2) = 1

k = 1

Putting this value in equation (ii) we obtain,

AC = BC

Thus, ∠A = ∠B (angles opposite to equal sides of triangle are equal.)

☛ Check: NCERT Solutions Class 10 Maths Chapter 8Video Solution:

## If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B

Maths NCERT Solutions Class 10  Chapter 8 Exercise 8.1 Question 6

Summary:

If ∠A and ∠B are acute angles such that cos A = cos B we have proved that ∠A = ∠B since AC = BC and angles opposite to equal sides of a triangle are equal.

☛ Related Questions:

If cotθ = 7/8, evaluate: (i) (1 + sinθ)(1 - sinθ) / (1 + cosθ)(1 - cosθ), (ii) cot2θ

If 3 cot A = 4, check whether (1 - tan2 A) / (1 + tan2 A) = cos2 A - sin2 A or not.

In the triangle ABC right-angled at B, if tan A = 1/√3 find the value of:(i) sin A cos C + cos A sin C(ii) cos A cos C - sin A sin C

In ΔPQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.

स्रोत : www.cuemath.com

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Mohammed 2 month ago

Guys, does anyone know the answer?