Square Relation

Trigonometry of Class 10

THEOREM 2:

For any acute angle θ, prove that

(i) Square Relation;

(ii) Square Relation;

(iii) Square Relation.

Proof: Consider a right-angle ΔPQR in which ∠Q = 90º and ∠P = θº. Let PQ = x units, RQ = y units and PR = z units.

Then, by Pythagoras' theorem, we have

Square Relation

Square Relation

Now, (i)

Square Relation

[∴Square Relation]

Square Relation.

(ii) Square Relation  .

Square Relation.

(iii) Square Relation

 

[∴Square Relation]

Square Relation.

Q1. In the given triangle AB = 3 cm and AC = 5 cm. Find all trigonometric ratios.

Solution: Using Pythagoras theorem,

AC2 = AB2 + BC2

2.jpg

⇒ 52 = 32 + P2

⇒ 16 = P2  ⇒  P = 4 cm

Here P = 4 cm, B = 3 cm, H = 5 cm

Square Relation

question 2. If tan θ = m/n, then find sin θ.

Solution: Let P = mα and B = nα

3.jpg

Q3. If cosec A = 13/5 , the prove that tan2 A – sin2 A = sin4 A sec2 A.

Solution: We hare cosec A = 13/5 = Hypotenuse/Perpendicular.

So, we draw a right triangle ABC, right angled at C such that hypotenuse AB = 13 units and perpendicular

BC = 5 units.

By Pythagoras theorem,

4.jpg

AB2 = BC2 + AC2 Square Relation (13)2 = (5)2 + AC2

AC2 = 169 – 25 = 144

AC = √144 = 12 units

Square Relation

L.H.S. tan2 A – Sin2 A R.H.S. = sin4 A × sec2 A

Square Relation

So, L.H.S. = R.H.S. Hence, proved.

Q4. In ΔABC, right angled at B, AC + AB = 9 cm. Determine the value of cot C, cosec C, sec C.

Solution: In Δ ABC, we have

(AC)2 = (AB)2 + BC2

⇒ (9 – AB)2 = AB2 + (3)2 [ AC + AB = 9cm ⇒ AC = 9 – AB]

5.jpg

⇒ (81 + AB2 – 18AB = AB2 + 9

⇒ 72 – 18 AB = 0

⇒ AB = 72/18 = 4cm

Now, AC + AB = 9 cm

AC = 9 – 4 = 5 cm.

So, cot C = BC/AB = 3/4,cosec C = AC/AB = -5/4,sec C = AC/BC = 5/3.

Q5. Given that cos (A – B) = cos A cos B + sin A sin B, find the value of cos 15o.

Solution: Putting A = 45o and B = 30o, we get

we get cos (45o – 30o) = cos 45o cos 30o + sin 45o sin 30o

Square Relation

Q6. Evaluate:

(i) sin 60º cos 30º – cos 60º sin 30º

(ii) tan 30º cosec 60º + tan 60º sec 30º

Solution: (i) sin 60º cos 30º – cos 60º sin 30º

Square Relation.

(ii) tan 30º cosec 60º + tan 60º sec 30º

question 7. Evaluate:

(i) Square Relation

(ii) Square Relation

Solution:

(i) Square Relation

Square Relation

(ii) Square Relation

Square Relation.

Q8. In ΔPQS, right-angled at Q, PQ = 3 cm and PS = 6 cm. Determine ∠QPS and ∠PSQ.

Sol. Given PQ = 3 cm and PS = 6 cm.

Square Relation

or Square Relation.

So, ∠PSQ = 30º and ∴ ∠QPS = 60º.

Square Relation

Q9. If sin (A – B) = 1/2, cos (A + B) = 1/2, 0º < A + B ≤ 90º, A > B, find A and B.

Solution: Since, sin (A – B) = 1/2

or sin(A - B) = sin30°

therefore, A – B = 30º. ... (i)

Also, since cos (A + B) = 1/2

or cos(A + B) = cos60°

therefore, A + B = 60º. ... (ii)

Adding (i) and (ii), we have A = 45º and B = 15º.

Frequently Asked Question

Q1. What is Square relation in trigonometry?

Ans. Square relations among various trigonometrical ratios

sin2θ+cos2θ=1. sec2θ−tan2θ=1. cosec2θ−cot2θ=1. These are also called Pythagorean Identities.

Q2. What are the formulas of trigonometry class 10?

Ans. Class 10 Trigonometry Formulas

sin(90° - A) = cos A

cos(90° - A) = sin A

tan(90° - A) = cot A

cot(90° - A) = tan A

sec(90° - A) = cosec A

cosec(90° - A) = sec A

sin2 θ + cos2 θ = 1 ⇒ sin2 θ = 1 - cos2 θ ⇒ cos2 θ = 1 - sin2 θ

cosec2 θ - cot2 θ = 1 ⇒ cosec2 θ = 1 + cot2 θ ⇒ cot2 θ = cosec2 θ - 1.

Q3. How do you do Class 10 trigonometry easily?

Ans. 7 Easy Steps to Learn Trigonometry

  • Study all the basics of trigonometric angles.
  • Study right-angle triangle concepts.
  • Pythagoras theorem.
  • Sine rule and Cosine rule.
  • List all the important identities of trigonometry.
  • Remember the trigonometry table.
  • Be thorough with the trigonometric formulas.

Q4. Is trigonometry important for JEE?

Ans. Trigonometry: It is one of the most important topics for JEE Main. Inverse Trigonometry, Trigonometric functions, trigonometric ratios and trigonometric identities are some of the most imperative topics where you need to focus on.

Q5. Why is trigonometry so hard?

Ans. With trigonometry, you have to remember what sine and cosine mean, for example. You have to remember what they represent and the various ways they impact angles and lengths. Trigonometry is difficult because it involves a lot of memorization of different functions which can then deviate into other functions.

Q6. What is relation between Sinθ and Cosecθ?

Ans. Reciprocal Relationship

As the name suggests, these relations involve two trigonometric ratios which are connected by inverse relations between them. For example, sin θ = 1/ cosec θ or sin θ x cosec θ = 1. cos θ = 1/ sec θ or cos θ x sec θ = 1.

Q7. What is the relation between tanθ and cotθ?

Ans. The Tanθ is the ratio of the altitude, base of the right triangle, and Cotθ is the ratio of the base, altitude of the right triangle. The cot - tan formula presents an inverse relationship between Cotθ and Tanθ.

Also Check

  1. Quotient Relation Of Trigonometry Ratios
  2. Trigonometric Ratios Of Complementary Angles
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