Introduction to Trigonometry · Class 10 Mathematics: 173 MCQs with answers

Q1. In a right triangle, the hypotenuse is the side

Q2. $\sin A$ is defined as

Q3. $\cos A$ is defined as

Q4. $\tan A$ is defined as

Q5. $\operatorname{cosec} A$ is the reciprocal of

Q6. $\sec A$ is the reciprocal of

Q7. $\cot A$ is the reciprocal of

Q8. In a triangle with opposite 3, adjacent 4 and hypotenuse 5, $\sin A$ is

Q9. In the same triangle, $\cos A$ is

Q10. In the same triangle, $\tan A$ is

Q11. $\tan A$ can also be written as

Q12. Assertion (A): Two right triangles with the same acute angle give the same value of $\sin$ for that angle. Reason (R): The two triangles have the same area.

Q13. Which side is never affected by which acute angle you choose?

Q14. In a triangle with opposite 5, adjacent 12 and hypotenuse 13, $\sin A$ is

Q15. SOH-CAH-TOA is a way to remember

Q16. In a right triangle, define $\sin A$, $\cos A$ and $\tan A$ in terms of the sides.

Q17. State the three reciprocal ratios, saying which ratio each one inverts.

Q18. $\sin 30°$ equals

Q19. $\cos 60°$ equals

Q20. $\sin 45°$ equals

Q21. $\tan 45°$ equals

Q22. $\cos 0°$ equals

Q23. $\sin 0°$ equals

Q24. $\sin 90°$ equals

Q25. $\cos 90°$ equals

Q26. $\tan 90°$ is

Q27. $\tan 60°$ equals

Q28. $\tan 30°$ equals

Q29. Assertion (A): $\sin 45° = \cos 45°$. Reason (R): The 45° values come from a right triangle whose two legs are equal.

Q30. $\cos 30°$ equals

Q31. $\sin 45°$ and $\cos 45°$ are

Q32. $\sec^2 45°$ equals

Q33. Assertion (A): The 30° and 60° values can both be read from one triangle. Reason (R): Bisecting an equilateral triangle produces a right triangle containing both a 30° and a 60° angle.

Q34. Write the values of $\sin$, $\cos$ and $\tan$ for 30°, 45° and 60°.

Q35. Find the acute angle $\theta$ for which $\tan\theta = \sqrt3$.

Q36. $\sin^2\theta + \cos^2\theta$ equals

Q37. $\sec^2\theta - \tan^2\theta$ equals

Q38. $\operatorname{cosec}^2\theta - \cot^2\theta$ equals

Q39. Identity 1 is proved from

Q40. To obtain $\sec^2\theta - \tan^2\theta = 1$, divide identity 1 by

Q41. To obtain $\operatorname{cosec}^2\theta - \cot^2\theta = 1$, divide identity 1 by

Q42. $1 - \cos^2\theta$ equals

Q43. $1 + \tan^2\theta$ equals

Q44. $1 + \cot^2\theta$ equals

Q45. An identity is a statement that is true

Q46. When proving an identity you should work on

Q47. Assertion (A): $\sin^2\theta + \cos^2\theta = 1$ is an identity. Reason (R): It is true for every value of $\theta$, not merely for particular ones.

Q48. $\sec^2\theta - 1$ equals

Q49. $\operatorname{cosec}^2\theta - 1$ equals

Q50. $\frac{\tan\theta}{\sin\theta}$ equals

Q51. State the three trigonometric identities.

Q52. Which of these is impossible?

Q53. The smallest possible value of $\sec A$ for an acute angle is

Q54. Which pair of ratios has NO upper limit?

Q55. $\sin^2 A$ means

Q56. $\sin^{-1} A$ means

Q57. Taking the hypotenuse as 5 when $\sin A = \frac35$, the adjacent side is

Q58. A right triangle has $\sin A = \frac35$. Its cosine is

Q59. With $\sin A = \frac35$, the value of $\cot A$ is

Q60. If $\cos A = \frac{12}{13}$, then $\sin A$ is

Q61. If $\tan A = \frac12$, the hypotenuse of the simplest such triangle is

Q62. Why may you take the sides as 1 and 2 when $\tan A = \frac12$?

Q63. Assertion (A): $\sin^{-1} A$ is the same as $\operatorname{cosec} A$. Reason (R): $\operatorname{cosec} A$ is defined as $\frac{1}{\sin A}$.

Q64. Assertion (A): When the acute angle is switched to the other one, the hypotenuse changes. Reason (R): 'Opposite' and 'adjacent' are named relative to the chosen angle.

Q65. In that same triangle, $\sec A$ is

Q66. A triangle has $\sin A = \frac{7}{25}$. Its cosine is

Q67. $\cot A$ can also be written as

Q68. Assertion (A): Given only $\cos A$, every other ratio of A can be found. Reason (R): The given ratio fixes two sides, and Pythagoras supplies the third.

Q69. Explain why the value of $\sin A$ does not depend on the size of the right triangle.

Q70. Given $\sin A = \frac{3}{5}$, find all the other five ratios.

Q71. Given $\cos A = \frac{12}{13}$, find $\sin A$ and $\tan A$.

Q72. Given $\tan A = \frac{1}{2}$, find $\sin A$ and $\cos A$.

Q73. Explain why $\sin A$ can never be greater than 1.

Q74. Explain why $\sec A$ can never be less than 1.

Q75. Show that $\tan A \times \cot A = 1$ for any acute angle A.

Q76. (a) In a right triangle ABC right-angled at B, explain how the three sides are named relative to angle A. (b) Define all six trigonometric ratios of A. (c) If $AB = 24$ cm, $BC = 7$ cm, find all six ratios of angle A.

Q77. The 45° ratios are derived from

Q78. The 30° and 60° ratios are derived from

Q79. Cutting an equilateral triangle of side 2 in half gives sides

Q80. As the angle grows from 0° to 90°, $\cos$

Q81. The cosine row of the table is

Q82. $\operatorname{cosec} 30°$ equals

Q83. $\sec 60°$ equals

Q84. $\cot 45°$ equals

Q85. $\operatorname{cosec} 0°$ is

Q86. $\operatorname{cosec} 30° + \sec 60°$ equals

Q87. $\sin^2 60° + \cos^2 30°$ equals

Q88. Assertion (A): $\tan 90°$ equals infinity. Reason (R): The adjacent side becomes zero as the angle approaches 90°.

Q89. Assertion (A): $\sin 60^\circ = \cos 60^\circ$. Reason (R): The cosine row of the table is the sine row read backwards.

Q90. $\cot 30°$ equals

Q91. $\sec 45°$ equals

Q92. $\frac{\sin 60°}{\cos 60°}$ equals

Q93. As the acute angle increases, $\tan$

Q94. Derive the values of $\sin 45°$, $\cos 45°$ and $\tan 45°$.

Q95. Explain why $\tan 90°$ is undefined.

Q96. Evaluate $\sin 30° \cos 60° + \cos 30° \sin 60°$.

Q97. Evaluate $2 \tan^2 45° + \cos^2 30° - \sin^2 60°$.

Q98. Evaluate $\cos 60° \cos 30° - \sin 60° \sin 30°$.

Q99. Evaluate $\frac{\tan 30°}{\tan 60°}$.

Q100. (a) Write out the complete table of $\sin$, $\cos$ and $\tan$ for 0°, 30°, 45°, 60° and 90°. (b) State two patterns in the table that reduce what must be memorised. (c) Use the table to evaluate $\sin^2 30° + \cos^2 30°$ and $\sin^2 45° + \cos^2 45°$, and comment.

Q101. Does $\sin^2\theta + \cos^2\theta = 1$ imply $\sin\theta + \cos\theta = 1$?

Q102. $\frac{\sec\theta}{\operatorname{cosec}\theta}$ equals

Q103. $\sin^4\theta - \cos^4\theta$ simplifies to

Q104. $\frac{1}{\sec\theta - \tan\theta}$ simplifies to

Q105. The conjugate of $\operatorname{cosec}\theta + \cot\theta$ is

Q106. When multiplying by a conjugate you must multiply

Q107. $\cot\theta \times \sin\theta$ equals

Q108. $\tan\theta + \cot\theta$ equals

Q109. The most reliable move when a proof stalls is to

Q110. Identity 2 is NOT valid at

Q111. Identity 3 is NOT valid at

Q112. Assertion (A): All three identities are really the same fact. Reason (R): Identities 2 and 3 are obtained from identity 1 by division alone.

Q113. Assertion (A): An identity can be solved to find $\theta$. Reason (R): An identity holds for every value of $\theta$.

Q114. $\cos\theta \times \operatorname{cosec}\theta$ equals

Q115. $\cos^2\theta \sec^2\theta + \sin^2\theta \operatorname{cosec}^2\theta$ equals

Q116. $\tan\theta \cos\theta$ equals

Q117. Assertion (A): Identity 1 can be rearranged into six different useful forms. Reason (R): Each identity can be written with either squared term isolated on one side.

Q118. Prove that $\sin^2\theta + \cos^2\theta = 1$.

Q119. Derive $\sec^2\theta - \tan^2\theta = 1$ from the first identity.

Q120. Explain the difference between a trigonometric identity and a trigonometric equation.

Q121. Prove that $(1 - \cos^2\theta)\operatorname{cosec}^2\theta = 1$.

Q122. Simplify $\sin^4\theta - \cos^4\theta$.

Q123. If $\cot A = \frac{7}{8}$, then $\operatorname{cosec} A$ is

Q124. Where $\sin A = \frac{8}{17}$, the tangent of A is

Q125. If $15\cot A = 8$, then $\sin A$ is

Q126. The product $\sin A \times \operatorname{cosec} A$ equals

Q127. The product $\tan A \times \cot A$ equals

Q128. If $\sin A = \cos A$ for an acute angle A, the triangle is

Q129. In a right triangle, $\frac{\sin A}{\cos A} \times \frac{\cos A}{\sin A}$ equals

Q130. Which statement about an acute angle A is FALSE?

Q131. Assertion (A): Knowing one trigonometric ratio determines all the others. Reason (R): The given ratio fixes two sides of a right triangle, and Pythagoras then fixes the third.

Q132. If the sides of a right triangle are doubled, the value of $\tan A$

Q133. Explain the difference between $\sin^{-1} A$ and $\operatorname{cosec} A$.

Q134. If $15 \cot A = 8$, find $\sin A$ and $\sec A$.

Q135. (a) Explain why the trigonometric ratios of an angle do not depend on the size of the triangle. (b) Hence explain why, given $\tan A = \frac34$, you may take the opposite side as 3 and the adjacent as 4. (c) Using those sides, find $\sin A$, $\cos A$ and $\operatorname{cosec} A$.

Q136. (a) State the limits on the values of $\sin A$, $\cos A$, $\sec A$ and $\operatorname{cosec} A$ for an acute angle, giving a reason for each. (b) Explain why $\tan A$ has no upper limit. (c) A student writes $\sin A = \frac{7}{5}$. Explain what is wrong and how it could be spotted immediately.

Q137. In a right triangle PQR, right-angled at Q, $PQ = 5$ cm and $PR = 13$ cm. (a) Find QR. (b) Find $\sin P$, $\cos P$ and $\tan P$. (c) Find $\sin R$ and $\cos R$, and comment on what you notice.

Q138. $2 \tan^2 45° + \cos^2 30° - \sin^2 60°$ equals

Q139. $\sin(30° + 60°)$ equals

Q140. Which statement is true?

Q141. $\frac{\tan 30°}{\tan 60°}$ equals

Q142. $\tan 60° - \cot 30°$ equals

Q143. $2 \sin 30° \cos 30°$ equals

Q144. If $\sin \theta = \frac{\sqrt3}{2}$ for an acute angle, then $\theta$ is

Q145. If $\tan \theta = 1$ for an acute angle, then $\theta$ is

Q146. Assertion (A): Every value in the standard table can be obtained from just two triangles. Reason (R): Half a square gives the $30^\circ$ and $60^\circ$ values, and half an equilateral triangle gives the $45^\circ$ values.

Q147. $\tan 45° \div \tan 60°$ equals

Q148. Derive the values of $\sin 30°$ and $\cos 30°$.

Q149. Show that $\sin(30° + 60°) \ne \sin 30° + \sin 60°$.

Q150. List the standard values that are undefined, with a reason.

Q151. (a) Draw and describe the triangle used to obtain the 45° ratios, and state all three. (b) Do the same for the 30° and 60° ratios. (c) Explain why the 0° and 90° values cannot be obtained from a drawn triangle.

Q152. (a) Evaluate $\frac{5 \cos^2 60° + 4 \sec^2 30° - \tan^2 45°}{\sin^2 30° + \cos^2 30°}$. (b) Explain why the denominator could have been written down without calculation. (c) State which standard values are undefined and why.

Q153. A surveyor uses standard angles for quick estimates. (a) A ramp makes 30° with the ground and is 20 m long. Find its vertical rise. (b) A roof makes 45° with the horizontal and rises 4 m. Find its horizontal run. (c) A wire makes 60° with the ground and is anchored 5 m from the base of a pole. Find the length of the wire.

Q154. $\frac{1 - \cos\theta}{\sin\theta}$ equals

Q155. $\sec^2\theta + \operatorname{cosec}^2\theta$ equals

Q156. $1 - 2\sin^2\theta$ equals

Q157. $\sin^4\theta + 2\sin^2\theta\cos^2\theta + \cos^4\theta - 1$ equals

Q158. If $\sin\theta = \frac35$, then $\cos^2\theta$ equals

Q159. $\frac{\cos^2\theta}{1 - \sin\theta}$ equals

Q160. $\sqrt{\sec^2\theta - 1}$ for an acute angle equals

Q161. $\tan^2\theta - \sin^2\theta$ equals

Q162. Assertion (A): Multiplying by a conjugate does not change the value of an expression. Reason (R): The conjugate is used as a fraction over itself, which equals 1.

Q163. Which technique suits $\sin^4\theta - \cos^4\theta$?

Q164. Identity 1 says the point $(\cos\theta, \sin\theta)$ always lies on

Q165. Prove that $\tan\theta + \cot\theta = \sec\theta \operatorname{cosec}\theta$.

Q166. Show that $\frac{1}{\sec\theta - \tan\theta} = \sec\theta + \tan\theta$.

Q167. Prove that $(\sin\theta + \cos\theta)^2 + (\sin\theta - \cos\theta)^2 = 2$.

Q168. Simplify $\frac{\cos^2\theta}{1 - \sin\theta}$.

Q169. Explain why you must work on only one side when proving an identity.

Q170. (a) Prove that $\sin^2\theta + \cos^2\theta = 1$, stating clearly what you use. (b) Derive the other two identities from it. (c) State the range of angles for which each of the three is valid, with a reason.

Q171. (a) State the rule for proving a trigonometric identity, and why it matters. (b) Prove that $\frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta}$. (c) Name the technique you used and explain why it works.

Q172. (a) Prove that $\tan\theta + \cot\theta = \sec\theta \operatorname{cosec}\theta$. (b) Prove that $\sec^2\theta + \operatorname{cosec}^2\theta = \sec^2\theta \operatorname{cosec}^2\theta$. (c) State the single technique both proofs share.

Q173. (a) Given that $\sin\theta = \frac{3}{5}$ for an acute angle, find $\cos\theta$ and $\tan\theta$ using an identity rather than a triangle. (b) Verify your answer satisfies $\sec^2\theta - \tan^2\theta = 1$. (c) Explain why using the identity gives the same answer as drawing a triangle.