Trigonometry is the study of relationships between the sides and angles of a right angled triangle.
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:
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
Here (90° – A) is the complementary angle of A.
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) sin2θ + cos2θ = 1 [for 0° ≤ θ ≤ 90°]
(ii) sec2θ – tan2θ = 1 [for 0° ≤ θ ≤ 90°]
(iii) cosec2θ – cot2θ = 1 [for 0° < θ ≤ 90°]
In ∆ABC right angled at B, AB = 24 cm, BC = 7 cm. Determine:
(i) sin A, cos A
(ii) sin C, cos C
In given figure, find tan P – cot R.
If sin A = , calculate cos A and tan A.
Given 15 cot A = 8, find sin A and sec A.
Given sec θ = , calculate all other trigonometric ratios.
If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B.
If cot θ = , evaluate:
If 3 cot A = 4, check whether = cos² A – sin² A or not.
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
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.
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.