Class 10 › Mathematics Introduction to Trigonometry Class 10 Mathematics · Chapter 8
Practice all → Summary Questions & answers Practise this chapter → View all Page 1 free about 8 min read , Full kit about 55 min read Page 1 free about 8 min read , Full kit about 55 min read View all 12 amazing Marks Booster kits → Trigonometry studies the relationships between the angles and sides of right-angled triangles. The chapter defines the trigonometric ratios — sine, cosine, tangent and their reciprocals — for an acute angle, and gives their values for the standard angles \(0^\circ, 30^\circ, 45^\circ, 60^\circ\) and \(90^\circ\). It also establishes fundamental trigonometric identities such as \(\sin^2\theta + \cos^2\theta = 1\), which are used to simplify expressions and prove results. These ratios and identities are the basis for solving a wide range of measurement problems.
Trigonometric ratios
\(\sin,\ \cos,\ \tan\) and their reciprocals for an acute angle.
Identity \(\sin^2\theta + \cos^2\theta = 1\)
A fundamental relation true for all angles.
Standard angles
Angles \(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\) with known ratio values.
Reciprocal ratios
\(\operatorname{cosec},\ \sec\) and \(\cot\) — reciprocals of \(\sin,\ \cos\) and \(\tan\).
Q1 Write the value of \(\sin 30^\circ\) and \(\cos 60^\circ\). Answer Q2 State the identity connecting \(\sin\theta\) and \(\cos\theta\). Answer Q3 Define the trigonometric ratios sine, cosine and tangent for an acute angle. Answer Q4 Write the values of \(\sin 30^\circ\), \(\cos 60^\circ\) and \(\tan 45^\circ\). Answer Q5 State the relationship between \(\tan\theta\), \(\sin\theta\) and \(\cos\theta\). Answer Q6 State the fundamental trigonometric identity. Answer Q7 Write the other two Pythagorean identities. Answer Q8 Name the reciprocal ratios of sin, cos and tan. Answer Q9 If \(\sin\theta = 3/5\), find \(\cos\theta\). Answer Q10 Evaluate \(\sin 30^\circ \cos 60^\circ + \cos 30^\circ \sin 60^\circ\). Answer Q1 1. In \(\triangle ABC\), right-angled at \(B\), \(AB = 24\text{ cm}\), \(BC = 7\text{ cm}\). Determine : (i) \(\sin A, \cos A\) Answer Q2 1. In \(\triangle ABC\), right-angled at \(B\), \(AB = 24\text{ cm}\), \(BC = 7\text{ cm}\). Determine : (ii) \(\sin C, \cos C\) Answer Q3 2. In Fig. 8.13, find \(\tan P-\cot R\). Answer Q4 3. If \(\sin A=\frac{3}{4}\), calculate \(\cos A\) and \(\tan A\). Answer Q5 4. Given \(15\cot A=8\), find \(\sin A\) and \(\sec A\). Answer Q6 5. Given \(\sec\theta=\frac{13}{12}\), calculate all other trigonometric ratios. Answer Q7 6. If \(\angle A\) and \(\angle B\) are acute angles such that \(\cos A=\cos B\), then show that \(\angle A=\angle B\). Answer Q8 7. If \(\cot\theta=\frac{7}{8}\), evaluate : (i) \(\frac{(1+\sin\theta)(1-\sin\theta)}{(1+\cos\theta)(1-\cos\theta)}\) Answer Q9 7. If \(\cot\theta=\frac{7}{8}\), evaluate : (ii) \(\cot^2\theta\) Answer Q10 8. If \(3\cot A=4\), check whether \(\frac{1-\tan^2 A}{1+\tan^2 A}=\cos^2 A-\sin^2 A\) or not. Answer Q11 9. In triangle \(ABC\), right-angled at \(B\), if \(\tan A=\frac{1}{\sqrt3}\), find the value of: (i) \(\sin A\cos C+\cos A\sin C\) Answer Q12 9. In triangle \(ABC\), right-angled at \(B\), if \(\tan A=\frac{1}{\sqrt3}\), find the value of: (ii) \(\cos A\cos C-\sin A\sin C\) Answer Q13 10. In \(\triangle PQR\), right-angled at \(Q\), \(PR+QR=25\text{ cm}\) and \(PQ=5\text{ cm}\). Determine the values of \(\sin P,\cos P\) and \(\tan P\). Answer Q14 11. State whether the following are true or false. Justify your answer. (i) The value of \(\tan A\) is always less than \(1\). Answer Q15 11. State whether the following are true or false. Justify your answer. (ii) \(\sec A=\frac{12}{5}\) for some value of angle \(A\). Answer Q16 11. State whether the following are true or false. Justify your answer. (iii) \(\cos A\) is the abbreviation used for the cosecant of angle \(A\). Answer Q17 11. State whether the following are true or false. Justify your answer. (iv) \(\cot A\) is the product of \(\cot\) and \(A\). Answer Q18 11. State whether the following are true or false. Justify your answer. (v) \(\sin\theta=\frac43\) for some angle \(\theta\). Answer Q1 1. Evaluate the following : (i) \(\sin60^\circ\cos30^\circ+\sin30^\circ\cos60^\circ\) Answer Q2 1. Evaluate the following : (ii) \(2\tan^2 45^\circ+\cos^2 30^\circ-\sin^2 60^\circ\) Answer Q3 1. Evaluate the following : (iii) \(\frac{\cos45^\circ}{\sec30^\circ+\cosec30^\circ}\) Answer Q4 1. Evaluate the following : (iv) \(\frac{\sin30^\circ+\tan45^\circ-\cosec60^\circ}{\sec30^\circ+\cos60^\circ+\cot45^\circ}\) Answer Q5 1. Evaluate the following : (v) \(\frac{5\cos^2 60^\circ+4\sec^2 30^\circ-\tan^2 45^\circ}{\sin^2 30^\circ+\cos^2 30^\circ}\) Answer Q6 2. Choose the correct option and justify your choice : (i) \(\frac{2\tan30^\circ}{1+\tan^2 30^\circ}=\) (A) \(\sin60^\circ\) (B) \(\cos60^\circ\) (C) \(\tan60^\circ\) (D) \(\sin30^\circ\) Answer Q7 2. Choose the correct option and justify your choice : (ii) \(\frac{1-\tan^2 45^\circ}{1+\tan^2 45^\circ}=\) (A) \(\tan90^\circ\) (B) \(1\) (C) \(\sin45^\circ\) (D) \(0\) Answer Q8 2. Choose the correct option and justify your choice : (iii) \(\sin2A=2\sin A\) is true when \(A=\) (A) \(0^\circ\) (B) \(30^\circ\) (C) \(45^\circ\) (D) \(60^\circ\) Answer Q9 2. Choose the correct option and justify your choice : (iv) \(\frac{2\tan30^\circ}{1-\tan^2 30^\circ}=\) (A) \(\cos60^\circ\) (B) \(\sin60^\circ\) (C) \(\tan60^\circ\) (D) \(\sin30^\circ\) Answer Q10 3. If \(\tan(A+B)=\sqrt3\) and \(\tan(A-B)=\frac1{\sqrt3}\); \(0^\circB\), find \(A\) and \(B\). Answer Q11 4. State whether the following are true or false. Justify your answer. (i) \(\sin(A+B)=\sin A+\sin B\). Answer Q12 4. State whether the following are true or false. Justify your answer. (ii) The value of \(\sin\theta\) increases as \(\theta\) increases. Answer Q13 4. State whether the following are true or false. Justify your answer. (iii) The value of \(\cos\theta\) increases as \(\theta\) increases. Answer Q14 4. State whether the following are true or false. Justify your answer. (iv) \(\sin\theta=\cos\theta\) for all values of \(\theta\). Answer Q15 4. State whether the following are true or false. Justify your answer. (v) \(\cot A\) is not defined for \(A=0^\circ\). Answer Q1 1. Express the trigonometric ratios \(\sin A,\sec A\) and \(\tan A\) in terms of \(\cot A\). Answer Q2 2. Write all the other trigonometric ratios of \(\angle A\) in terms of \(\sec A\). Answer Q3 3. Choose the correct option. Justify your choice. (i) \(9\sec^2 A-9\tan^2 A=\) (A) \(1\) (B) \(9\) (C) \(8\) (D) \(0\) Answer Q4 3. Choose the correct option. Justify your choice. (ii) \((1+\tan\theta+\sec\theta)(1+\cot\theta-\cosec\theta)=\) (A) \(0\) (B) \(1\) (C) \(2\) (D) \(-1\) Answer Q5 3. Choose the correct option. Justify your choice. (iii) \((\sec A+\tan A)(1-\sin A)=\) (A) \(\sec A\) (B) \(\sin A\) (C) \(\cosec A\) (D) \(\cos A\) Answer Q6 3. Choose the correct option. Justify your choice. (iv) \(\frac{1+\tan^2 A}{1+\cot^2 A}=\) (A) \(\sec^2 A\) (B) \(-1\) (C) \(\cot^2 A\) (D) \(\tan^2 A\) Answer Q7 4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (i) \((\cosec\theta-\cot\theta)^2=\frac{1-\cos\theta}{1+\cos\theta}\) Answer Q8 4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (ii) \(\frac{\cos A}{1+\sin A}+\frac{1+\sin A}{\cos A}=2\sec A\) Answer Q9 4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (iii) \(\frac{\tan\theta}{1-\cot\theta}+\frac{\cot\theta}{1-\tan\theta}=1+\sec\theta\cosec\theta\) [Hint : Write the expression in terms of \(\sin\theta\) and \(\cos\theta\)] Answer Q10 4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (iv) \(\frac{1+\sec A}{\sec A}=\frac{\sin^2 A}{1-\cos A}\) [Hint : Simplify LHS and RHS separately] Answer Q11 4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (v) \(\frac{\cos A-\sin A+1}{\cos A+\sin A-1}=\cosec A+\cot A\), using the identity \(\cosec^2 A=1+\cot^2 A\). Answer Q12 4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (vi) \(\sqrt{\frac{1+\sin A}{1-\sin A}}=\sec A+\tan A\) Answer Q13 4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (vii) \(\frac{\sin\theta-2\sin^3\theta}{2\cos^3\theta-\cos\theta}=\tan\theta\) Answer Q14 4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (viii) \((\sin A+\cosec A)^2+(\cos A+\sec A)^2=7+\tan^2 A+\cot^2 A\) Answer Q15 4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (ix) \((\cosec A-\sin A)(\sec A-\cos A)=\frac1{\tan A+\cot A}\) [Hint : Simplify LHS and RHS separately] Answer Q16 4. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (x) \(\frac{1+\tan^2 A}{1+\cot^2 A}=\left(\frac{1-\tan A}{1-\cot A}\right)^2=\tan^2 A\) Answer
In a right triangle, sin θ is defined as:
A opposite / hypotenuse B opposite / adjacent C adjacent / hypotenuse D hypotenuse / opposite
Flashcards & interactive practice 
