CBSE Class 10 Mathematics Introduction to Trigonometry MCQs Set 09

Question. If \((\sec A + \tan A)(\sec B + \tan B)(\sec C + \tan C) = (\sec A - \tan A)(\sec B - \tan B)(\sec C - \tan C) = x\) then the value/values of \(x\) is/are
(a) \(\pm 1\)
(b) 0
(c) \(\pm 2\)
(d) 1
Answer: A
Multiplying both sides by \((\sec A - \tan A)(\sec B - \tan B)(\sec C - \tan C)\), we get
\((\sec^2 A - \tan^2 A)(\sec^2 B - \tan^2 B)(\sec^2 C - \tan^2 C) = x^2\)
\(1 \times 1 \times 1 = x^2\)
\(x = \pm 1\)

Question. If \(\sin\theta + \sin^2\theta = 1\), then find the value of \(\cos^{12}\theta + 3\cos^{10}\theta + 3\cos^8\theta + \cos^6\theta + 2\cos^4\theta + 2\cos^2\theta - 2\).
(a) 0
(b) 1
(c) \(\cos\theta\)
(d) \(\sin\theta\)
Answer: B
We have, \(\sin\theta + \sin^2\theta = 1 \Rightarrow \sin\theta = 1 - \sin^2\theta \Rightarrow \sin\theta = \cos^2\theta\)
The expression is \((\cos^{12}\theta + 3\cos^{10}\theta + 3\cos^8\theta + \cos^6\theta) + 2(\cos^4\theta + \cos^2\theta) - 2\)
\(= (\cos^4\theta + \cos^2\theta)^3 + 2(\cos^4\theta + \cos^2\theta) - 2\)
\(= (\sin^2\theta + \cos^2\theta)^3 + 2(\sin^2\theta + \cos^2\theta) - 2\)
\(= (1)^3 + 2(1) - 2 = 1\)

Question. If \(0^\circ < x < 90^\circ\) and \(2\sin x + 15\cos^2 x = 7\), then find the value of \(\tan x\).
(a) 4/5
(b) 3/5
(c) 3/4
(d) 4/3
Answer: D
Given, \(2\sin x + 15(1 - \sin^2 x) = 7\)
\(2\sin x + 15 - 15\sin^2 x = 7\)
\(15\sin^2 x - 2\sin x - 8 = 0\)
Solving the quadratic for \(\sin x\), we get \(\sin x = 4/5\)
Then \(\cos x = \sqrt{1 - (4/5)^2} = 3/5\)
\(\tan x = \frac{4/5}{3/5} = 4/3\)

Question. If \(f(x) = \cos^2 x + \sec^2 x\), then \(f(x)\)
(a) \(\geq 1\)
(b) \(\leq 1\)
(c) \(\geq 2\)
(d) \(\leq 2\)
Answer: C
\(f(x) = (\cos x - \sec x)^2 + 2\)
Since the square of any expression is \(\geq 0\), \(f(x) \geq 2\).

Question. If \(ABC\) is a right angled triangle, then find the relation between \(\tan\left(\frac{A-B-C}{2}\right)\) and \(\tan\left(\frac{A+B-C}{2}\right)\)
(a) equal
(b) unequal
(c) sum of these equal to 1
(d) None of the above
Answer: B
In \(\triangle ABC\), \(A + B + C = 180^\circ\).
\(\tan\left(\frac{A-B-C}{2}\right) = \tan\left(\frac{A-(180-A)}{2}\right) = \tan(A - 90^\circ) = -\cot A\)
\(\tan\left(\frac{A+B-C}{2}\right) = \tan\left(\frac{180-C-C}{2}\right) = \tan(90^\circ - C) = \cot C\)
The values are unequal.

Question. If \(\sin\theta + \sin^2\theta + \sin^3\theta = 1\), then \(\cos^6\theta - 4\cos^4\theta + 8\cos^2\theta\) is equal to
(a) 1
(b) 2
(c) 3
(d) 4
Answer: D
\(\sin\theta(1 + \sin^2\theta) = 1 - \sin^2\theta = \cos^2\theta\)
\(\sin\theta(2 - \cos^2\theta) = \cos^2\theta\)
Squaring: \((1 - \cos^2\theta)(4 + \cos^4\theta - 4\cos^2\theta) = \cos^4\theta\)
Simplifying leads to \(\cos^6\theta - 4\cos^4\theta + 8\cos^2\theta = 4\)

Question. If \(m = a\cos^3\theta + 3a\cos\theta\sin^2\theta\) and \(n = a\sin^3\theta + 3a\cos^2\theta\sin\theta\), then \((m + n)^{2/3} + (m - n)^{2/3}\) is equal to
(a) \(2a^{2/3}\)
(b) \(a^{2/3}\)
(c) \(2a^{3/2}\)
(d) \(a^{3/2}\)
Answer: A
\(m + n = a(\cos\theta + \sin\theta)^3 \Rightarrow (m + n)^{2/3} = a^{2/3}(\cos\theta + \sin\theta)^2\)
\(m - n = a(\cos\theta - \sin\theta)^3 \Rightarrow (m - n)^{2/3} = a^{2/3}(\cos\theta - \sin\theta)^2\)
Adding gives \(a^{2/3}[(\cos\theta + \sin\theta)^2 + (\cos\theta - \sin\theta)^2] = a^{2/3}[2(\cos^2\theta + \sin^2\theta)] = 2a^{2/3}\)

Question. If \(\tan \theta = \frac{a \sin \phi}{1 - a \cos \phi}\) and \(\tan \phi = \frac{b \sin \theta}{1 - b \cos \theta}\), then \(\frac{a}{b} =\)
(a) \(\frac{\sin \theta}{1 - \cos \theta}\)
(b) \(\frac{\sin \theta}{1 - \cos \phi}\)
(c) \(\frac{\sin \phi}{\sin \theta}\)
(d) \(\frac{\sin \theta}{\sin \phi}\)
Answer: D
We have, \(\tan \theta = \frac{a \sin \phi}{1 - a \cos \phi}\) \(\implies \cot \theta = \frac{1 - a \cos \phi}{a \sin \phi} = \frac{1}{a \sin \phi} - \cot \phi \implies \cot \theta + \cot \phi = \frac{1}{a \sin \phi}\) ...(1). Also, \(\tan \phi = \frac{b \sin \theta}{1 - b \cos \theta} \implies \cot \phi = \frac{1 - b \cos \theta}{b \sin \theta} = \frac{1}{b \sin \theta} - \cot \theta \implies \cot \phi + \cot \theta = \frac{1}{b \sin \theta}\) ...(2). From (1) and (2), we have \(\frac{1}{a \sin \phi} = \frac{1}{b \sin \theta} \implies \frac{a}{b} = \frac{\sin \theta}{\sin \phi}\)

Question. If \(a \sec \theta + b \tan \theta + c = 0\) and \(p \sec \theta + q \tan \theta + r = 0\), then \((br - qc)^2 - (pc - ar)^2\) is equal to
(a) \((ap - bq)^2\)
(b) \((aq - bp)^2\)
(c) \((ap - bq)\)
(d) \((aq - bp)\)
Answer: B
We have, \(a \sec \theta + b \tan \theta + c = 0\) and \(p \sec \theta + q \tan \theta + r = 0\). Solving these two equations for \(\sec \theta\) and \(\tan \theta\) by the cross-multiplication method, we get \(\frac{\sec \theta}{br - qc} = \frac{\tan \theta}{cp - ar} = \frac{1}{aq - bp}\). So, \(\sec \theta = \frac{br - cq}{aq - bp}\) and \(\tan \theta = \frac{cp - ar}{aq - bp}\). Now, \(\sec^2 \theta - \tan^2 \theta = 1 \implies \left(\frac{br - cq}{aq - bp}\right)^2 - \left(\frac{cp - ar}{aq - bp}\right)^2 = 1 \implies (br - cq)^2 - (cp - ar)^2 = (aq - bp)^2\)

FILL IN THE BLANK

Question. sine of \((90^\circ - \theta)\) is \(\dots\dots\dots\dots\)
Answer: \(\cos \theta\)

Question. \(\sin^2 A + \cos^2 A = \dots\dots\dots\dots\)
Answer: 1

Question. If \(\tan A = 4/3\) then \(\sin A\) \(\dots\dots\dots\dots\)
Answer: 4/5

Question. In a right triangle \(ABC\), right angled at \(B\), if \(\tan A = 1\), \(\sin A \cos A = \dots\dots\dots\dots\)
Answer: \(\frac{1}{2}\)

Question. Reciprocal of \(\sin \theta\) is \(\dots\dots\dots\dots\)
Answer: \(\csc \theta\)

Question. In \(\triangle ABC\), right-angled at \(B\), \(AB = 24 \text{ cm}\), \(BC = 7 \text{ cm}\). \(\sin A = \dots\dots\dots\dots\)
Answer: 7/25

Question. Maximum value for sine of any angle is \(\dots\dots\dots\dots\)
Answer: 1

Question. In \(\triangle PQR\), right-angled at \(Q\), \(PR + QR = 25 \text{ cm}\) and \(PQ = 5 \text{ cm}\). The value of \(\tan P\) is \(\dots\dots\dots\dots\)
Answer: 12/5

Question. Sum of \(\dots\dots\dots\dots\) of sine and cosine of angle is one.
Answer: Square

TRUE/FALSE

Question. \(\sin \theta = \cos \theta\) for all values of \(\theta\).
Answer: False

Question. The value of the expression \((\sin 80^\circ - \cos 80^\circ)\) is negative.
Answer: False

Question. \(\tan 48^\circ \tan 23^\circ \tan 42^\circ \tan 67^\circ \neq 1\)
Answer: False

Question. Trigonometry deals with measurement of components of triangles.
Answer: True

Question. \(\sin^2 \theta \times \cos^2 \theta = 1\)
Answer: False

Question. The value of \(\tan A\) is always less than 1.
Answer: False

Question. \(\frac{\tan 47^\circ}{\cot 43^\circ} = 1\)
Answer: True

Question. \(\sec A = 12/5\) for some value of angle \(A\).
Answer: True

Question. If \(\cos A + \cos^2 A = 1\), then \(\sin^2 A + \sin^4 A = 1\).
Answer: True

Question. \(\cos A\) is the abbreviation used for the cosecant of angle \(A\).
Answer: False

Question. \(\cot A\) is the product of \(\cot\) and \(A\).
Answer: False

Question. The value of \(\sin \theta + \cos \theta\) is always greater than 1.
Answer: False

Question. \(\sin \theta = \frac{4}{3}\) for some angle \(\theta\).
Answer: False

MATCHING QUESTIONS

Question. In \(\triangle ABC\), \(\angle B = 90^\circ\), \(AB = 3 \text{ cm}\) and \(BC = 4 \text{ cm}\) then match the column.
Column-I | Column-II
(A) \(\sin C\) | (p) 3/5
(B) \(\cos C\) | (q) 4/5
(C) \(\tan A\) | (r) 5/3
(D) \(\sec A\) | (s) 4/3
Answer: (A) - p, (B) - q, (C) - s, (D) - r

Question. Match the following:
Column-I | Column-II
(A) \(\frac{\cos A}{1 + \sin A} + \frac{1 + \sin A}{\cos A}\) | (p) \(\csc A + \cot A\)
(B) \(\frac{\cos A - \sin A + 1}{\cos A + \sin A - 1}\) | (q) \(2 \sec A\)
(C) \(\sqrt{\frac{1 + \sin A}{1 - \sin A}}\) | (r) \(\sec A + \tan A\)
(D) \(\frac{\sin^2 A}{1 - \cos A}\) | (s) \(\frac{1 + \sec A}{\sec A}\)
Answer: (A) - q, (B) - p, (C) - r, (D) - s

Question. If \(\sin A = \frac{7}{25}\), then match the following:
Column-I | Column-II
(A) \(\cos A\) | (p) 24/25
(B) \(\tan A\) | (q) 7/24
(C) \(\csc A\) | (r) 25/7
(D) \(\sec A\) | (s) 25/24
| (t) \(1 - 1/25\)
| (u) \(1 + 1/24\)
Answer: (A) - (p, t), (B) - q, (C) - r, (D) - (s, u)

Question. Match the following:
Column I | Column II
(A) \(\sin^2 37^\circ + \sin^2 53^\circ + \sin^2 90^\circ\) | (p) 0
(B) \(\tan 35^\circ \tan 45^\circ \tan 55^\circ\) | (q) 3
(C) \(\frac{\sec 72^\circ \sin 18^\circ + \tan 72^\circ \cot 18^\circ}{\cos 60^\circ}\) | (r) 1
(D) \(\frac{\tan 60^\circ}{\tan 30^\circ}\) | (s) 2
(E) \(\sin^2 30^\circ + \cos^2 30^\circ - \sin^2 60^\circ - \cos^2 60^\circ\) | (t) 4
Answer: (A) - s, (B) - r, (C) - t, (D) - q, (E) - p

ASSERTION AND REASON

Question. DIRECTION: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Question. Assertion: In a right angled triangle, if \(\tan \theta = \frac{3}{4}\), the greatest side of the triangle is 5 units.
Reason: \((\text{greatest side})^2 = (\text{hypotenuse})^2 = (\text{perpendicular})^2 + (\text{base})^2\).
(a) A
(b) B
(c) C
(d) D
Answer: A
Both Assertion and Reason are correct and Reason is the correct explanation of Assertion. Greatest side = \(\sqrt{(3)^2 + (4)^2} = 5\) units.

Question. Assertion: If \(\cos A + \cos^2 A = 1\) then \(\sin^2 A + \sin^4 A = 2\).
Reason: \(1 - \sin^2 A = \cos^2 A\), for any value of \(A\).
(a) A
(b) B
(c) C
(d) D
Answer: D
\(\cos A + \cos^2 A = 1 \implies \cos A = 1 - \cos^2 A = \sin^2 A\). So, \(\sin^2 A + \sin^4 A = \cos A + \cos^2 A = 1\). Assertion (A) is false but reason (R) is true.

Question. Assertion: In a right angled triangle, if \(\cos \theta = \frac{1}{2}\) and \(\sin \theta = \frac{\sqrt{3}}{2}\), then \(\tan \theta = \sqrt{3}\)
Reason: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
(a) A
(b) B
(c) C
(d) D
Answer: A
Both Assertion and Reason are correct and Reason is the correct explanation of Assertion. \(\tan \theta = \frac{\sqrt{3}/2}{1/2} = \sqrt{3}\).

Question. Assertion: The value of \(\sin \theta \cos(90^\circ - \theta) + \cos \theta \sin(90^\circ - \theta)\) equals to 1.
Reason: \(\tan \theta = \sec(90^\circ - \theta)\)
(a) A
(b) B
(c) C
(d) D
Answer: C
\(\sin \theta \cdot \sin \theta + \cos \theta \cdot \cos \theta = \sin^2 \theta + \cos^2 \theta = 1\). And, \(\tan \theta = \cot(90^\circ - \theta)\).