Introduction:

#### Trigonometry

Trigonometry is the study of relationships between the sides and angles of a right angled triangle.

#### Trigonometric Ratios

Trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the length of its sides.

Let ∆ABC be a triangle right angled at B. Then the trigonometric ratios of the angle A in right ∆ABC are defined as follows:

Note:

The values of the trigonometric ratios of an angle do not vary with the lengths of the sides of the triangle, if the angle remains same.

#### Trigonometric Ratios for Complementary Angles

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

Note:

Here (90° – A) is the complementary angle of A.

#### Trigonometric Identities

An equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of the angle(s) involved.

(i) sin^{2}θ + cos^{2}θ = 1 [for 0° ≤ θ ≤ 90°]

(ii) sec^{2}θ – tan^{2}θ = 1 [for 0° ≤ θ ≤ 90°]

(iii) cosec^{2}θ – cot^{2}θ = 1 [for 0° < θ ≤ 90°]

Question 1.

In ∆ABC right angled at B, AB = 24 cm, BC = 7 cm. Determine:

(i) sin A, cos A

(ii) sin C, cos C

Solution:

Question 2.

In given figure, find tan P – cot R.

Solution:

Question 3.

If sin A = , calculate cos A and tan A.

Solution:

Question 4.

Given 15 cot A = 8, find sin A and sec A.

Solution:

Question 5.

Given sec θ = , calculate all other trigonometric ratios.

Solution:

Question 6.

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

Solution:

Question 7.

If cot θ = , evaluate:

(i)

(ii) cot²θ

Question 8.

If 3 cot A = 4, check whether = cos² A – sin² A or not.

Question 9.

In triangle ABC, right angled at B, if tan A = , find the value of:

(i) sin A cos C + cos A sin C

(ii) cos A cos C – sin A sin C

Solution:

Question 10.

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.

Solution:

Question 11.

State whether the following statements are true or false. Justify your answer.

(i) The value of tan A is always less than 1.

(ii) sec A = for some value of angle A.

(iii) cos A is the abbreviation used for the cosecant of angle A.

(iv) cot A is the product of cot and A.

(v) sin θ = for some angle.

Solution:

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