CBSE Class 10 Maths HOTs Introduction to Trigonometry Set 07

Say True (T) or False (F).

Question. In a right angled triangle \( ABC \) right angled at \( B \) the value of \( \sin^2 A + \sin^2 C \) is 1.
Answer: T

Question. In the right angled triangle \( PQR \) right angled at \( Q \) if value of \( \tan^2 P \) is 1 then length \( PQ = \) length \( QR \).
Answer: T

Question. In a right angled triangle \( ABC \), right angled at \( B \), if \( \sin A = \cos C \) then triangle will be isosceles.
Answer: F

Question. Value of: \( \sin \theta + \cos \theta \) is always greater than or equal to one.
Answer: T

Question. In any \( \Delta ABC \), \( \sin\left(\frac{B+C}{2}\right) = \cos\left(\frac{A}{2}\right) \).
Answer: T

Question. The value of \( \cos 10^\circ \cdot \cos 20^\circ \cdot \cos 30^\circ \dots \cos 100^\circ = 0 \).
Answer: T

Question. Value of \( \tan \theta \) always lies between 1 and -1.
Answer: F

Question. If \( \cos \theta = x \), then \( -1 \le x \le 1 \).
Answer: T

Question. Value of : \( \sin 10^\circ \cdot \sin 20^\circ \cdot \sin 30^\circ \cdot \sec 70^\circ \cdot \sec 80^\circ = \frac{1}{2} \).
Answer: T

Question. \( \sec \theta = \frac{5}{12} \) is a possible relation.
Answer: F

Question. Value of \( \cot \theta \) is not defined for \( \theta = 90^\circ \).
Answer: F

Question. If \( H^2 = P^2 + B^2 \) is divided both sides by \( H^2 \) then trigonometric identity formed will be \( \sin^2 \theta + \cos^2 \theta = 1 \).
Answer: T

Question. Trigonometry is branch of geometry.
Answer: F

Question. Maximum value of \( \frac{1}{\text{cosec } \theta} \) is 1.
Answer: T

Question. Value of \( \sec \theta \) and \( \text{cosec } \theta \) can never be between - 1 and 1.
Answer: T

Fill in the Blank.

Question. Value of : \( \sec^2 \theta - \cot^2 (90^\circ - \theta) = \) ________.
Answer: 1

Question. Value of: \( \sin 12^\circ - \cos 78^\circ = \) ________.
Answer: 0

Question. Value of: \( \frac{\sec 41^\circ}{\text{cosec } 49^\circ} = \) ________.
Answer: 1

Question. \( \sin^2 \theta + \cos^2 \theta = \text{cosec}^2 \theta - \) ________.
Answer: \( \cot^2 \theta \)

Question. Value of \( \sin \theta \) can not be more than ________.
Answer: 1

Question. If \( \sec \theta + \tan \theta = p \) then value of \( \sec \theta - \tan \theta = \) ________.
Answer: \( \frac{1}{p} \)

Question. If \( \tan \theta + \cot \theta = 2 \) then value of \( \tan^2 \theta + \cot^2 \theta = \) ________.
Answer: 2

Question. If \( a \cos \theta + b \sin \theta = 4 \) and \( a \sin \theta - b \cos \theta = 3 \) then value of \( a^2 + b^2 \) is ________.
Answer: 25

Question. Value of \( \sec^2 \theta (1 + \sin \theta)(1 - \sin \theta) = \) ________.
Answer: 1

Question. Value of \( \sin^2 20^\circ + \sin^2 70^\circ = \) ________.
Answer: 1

Question. If \( \cos \theta + \cos^2 \theta = 1 \), then value of \( \sin^2 \theta + \sin^4 \theta \) is ________.
Answer: 1

Question. If \( x = a \sec \theta \cos \phi, y = b \sec \theta \sin \phi \) then value of \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = \) ________.
Answer: \( \sec^2 \theta \)

Question. If \( \sec \theta + \tan \theta = x \) then value of \( \sin \theta = \) ________.
Answer: \( \frac{x^2 - 1}{x^2 + 1} \)

Question. If \( \sin(\theta + 36^\circ) = \cos \theta \) then value of \( \theta \) is ________.
Answer: \( 27^\circ \)

Question. In a triangle \( ABC \) if \( A + B = 90^\circ \) then value of \( \cos C \) is ________.
Answer: 0

MCQs with more than one correct options.

Question. Value of \( \sin^2 \theta + \cos^2 \theta \) is same as value of
(a) \( \sin^2 \theta \cdot \cos^2 \theta \)
(b) \( \text{cosec}^2 \theta - \cot^2 \theta \)
(c) \( \sec^2 \theta - \tan^2 \theta \)
(d) \( \frac{1}{\sin^2 \theta} + \frac{1}{\cos^2 \theta} \)
Answer: (b) \( \text{cosec}^2 \theta - \cot^2 \theta \), (c) \( \sec^2 \theta - \tan^2 \theta \)

Question. Which of following has equal values
(a) \( \sin 30^\circ \)
(b) \( \tan 45^\circ \)
(c) \( \cos 60^\circ \)
(d) \( \tan 90^\circ \)
Answer: (a) \( \sin 30^\circ \), (c) \( \cos 60^\circ \)

Question. If \( \sec \theta = x + \frac{1}{4x} \), then value of \( \sec \theta + \tan \theta \) is equal to
(a) \( 2x \)
(b) \( \frac{2}{x} \)
(c) \( \frac{x}{2} \)
(d) \( \frac{1}{2x} \)
Answer: (a) \( 2x \), (d) \( \frac{1}{2x} \)

Question. If \( \sin \theta + \text{cosec } \theta = 2 \) then value of \( \sin^2 \theta + \text{cosec}^2 \theta \) is equal to
(a) 4
(b) 2
(c) \( \sin^4 \theta + \text{cosec}^4 \theta \)
(d) \( \frac{1}{\sin^4 \theta} + \frac{1}{\text{cosec}^4 \theta} \)
Answer: (b) 2, (c) \( \sin^4 \theta + \text{cosec}^4 \theta \), (d) \( \frac{1}{\sin^4 \theta} + \frac{1}{\text{cosec}^4 \theta} \)

Question. \( \cot A \) is also same as :
(a) \( \frac{\cos A}{\sin A} \)
(b) \( \sqrt{\text{cosec}^2 A - 1} \)
(c) \( \frac{\sin A}{\sec A} \)
(d) \( \frac{\text{cosec } A}{\sec A} \)
Answer: (a) \( \frac{\cos A}{\sin A} \), (b) \( \sqrt{\text{cosec}^2 A - 1} \), (d) \( \frac{\text{cosec } A}{\sec A} \)

Question. Which of the following are not possible values of \( \text{cosec } \theta \)
(a) \( \frac{1}{2} \)
(b) \( -\frac{1}{2} \)
(c) 3
(d) 4
Answer: (a) \( \frac{1}{2} \), (b) \( -\frac{1}{2} \)

Question. If \( y = (\sec A + \tan A)(\sec B + \tan B)(\sec C + \tan C) = (\sec A - \tan A)(\sec B - \tan B)(\sec C - \tan C) \) then value of \( y \) is
(a) -1
(b) 0
(c) 1
(d) 2
Answer: (a) -1, (c) 1

Question. In a right angled isosceles triangle \( ABC \) right angled at \( A \) which of the following trigonometric ratios have same value:
(a) \( \sin A \)
(b) \( \sin B \)
(c) \( \sin C \)
(d) \( \cos B \)
Answer: (b) \( \sin B \), (c) \( \sin C \), (d) \( \cos B \)

Question. Which of trigonometric ratios given below are equal
(a) \( \sin 30^\circ \)
(b) \( \cos 60^\circ \)
(c) \( \sin 90^\circ \)
(d) \( \cos 45^\circ \)
Answer: (a) \( \sin 30^\circ \), (b) \( \cos 60^\circ \)

Question. Value of \( \sin^2 27^\circ + \sin^2 63^\circ \) is same as value of
(a) \( \text{cosec}^2 10^\circ - \tan^2 80^\circ \)
(b) \( \sec^2 50^\circ - \cot^2 40^\circ \)
(c) \( \sin^2 10^\circ + \cos^2 70^\circ \)
(d) \( \sec^2 40^\circ - \cot^2 50^\circ \)
Answer: (a) \( \text{cosec}^2 10^\circ - \tan^2 80^\circ \), (b) \( \sec^2 50^\circ - \cot^2 40^\circ \)

Question. Which of following results are equal to 1
(a) \( \sin^3 A + \cos^3 A \)
(b) \( \frac{\sec^2 A}{1 + \tan^2 A} \)
(c) \( \frac{1 + \cot^2 A}{\text{cosec}^2 A} \)
(d) \( \sin^2 \theta + \cos^2 \phi \)
Answer: (b) \( \frac{\sec^2 A}{1 + \tan^2 A} \), (c) \( \frac{1 + \cot^2 A}{\text{cosec}^2 A} \)

Question. If \( \tan \theta = 1 \) then value of \( \sin 2\theta \) is equal to
(a) 1
(b) \( \cos 0^\circ \)
(c) \( \sin 90^\circ \)
(d) \( \cos 90^\circ \)
Answer: (a) 1, (b) \( \cos 0^\circ \), (c) \( \sin 90^\circ \)

Question. If \( \cos \theta = \frac{\sqrt{3}}{2} \) then which of following can not be value of \( \sin 2\theta \)
(a) 0
(b) \( \frac{\sqrt{3}}{2} \)
(c) \( -\frac{1}{2} \)
(d) 1
Answer: (a) 0, (c) \( -\frac{1}{2} \), (d) 1