✦ NCF 2023 ALIGNED • CBSE CLASS X ✦
Chapter 8: Introduction to Trigonometry
Assertion-Reason Question Bank
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Assertion (A)
The maximum value of \(\sin \theta\) for \(0^\circ \le \theta \le 90^\circ\) is \(1\).
Reason (R)
In a right triangle, perpendicular \(\le\) hypotenuse, so ratio \(\sin \theta = \text{Perpendicular}/\text{Hypotenuse} \le 1\), with maximum \(1\) attained at \(\theta = 90^\circ\).
Correct Option: (A)
Sine ratio represents perpendicular over hypotenuse, which cannot exceed \(1\). Reason explains Assertion.
Sine ratio represents perpendicular over hypotenuse, which cannot exceed \(1\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
For any angle \(\theta\), \(\sin \theta = 4/3\).
Reason (R)
Since perpendicular is always less than hypotenuse in a right triangle, \(\sin \theta \le 1\).
Correct Option: (D)
Assertion (A) is false (\(4/3 > 1\)). Reason (R) is true.
Assertion (A) is false (\(4/3 > 1\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The value of \(\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ\) is \(1\).
Reason (R)
Using trigonometric values: \(\left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{1}{2}\right)\left(\frac{1}{2}\right) = \frac{3}{4} + \frac{1}{4} = 1\).
Correct Option: (A)
Direct substitution of exact trigonometric values yields \(3/4 + 1/4 = 1\). Reason explains Assertion.
Direct substitution of exact trigonometric values yields \(3/4 + 1/4 = 1\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If \(\tan A = 4/3\), then \(\sin A = 4\) and \(\cos A = 3\).
Reason (R)
\(\sin A\) and \(\cos A\) can never exceed \(1\) for any real angle \(A\).
Correct Option: (D)
Assertion (A) is false (\(\sin A = 4/5, \cos A = 3/5\)). Reason (R) is true.
Assertion (A) is false (\(\sin A = 4/5, \cos A = 3/5\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The value of \(\sin 45^\circ\) is \(\frac{1}{\sqrt{2}}\).
Reason (R)
The value of \(\cos 45^\circ\) is \(\frac{1}{\sqrt{2}}\).
Correct Option: (B)
Both statements are true trigonometric values for \(45^\circ\).
Both statements are true trigonometric values for \(45^\circ\).
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
If \(\sin \theta = \cos \theta\) for an acute angle \(\theta\), then \(\theta = 45^\circ\).
Reason (R)
\(\tan 45^\circ = 1\).
Correct Option: (B)
Both A and R are true (\(\sin\theta/\cos\theta = 1 \implies \tan\theta = 1 \implies \theta = 45^\circ\)).
Both A and R are true (\(\sin\theta/\cos\theta = 1 \implies \tan\theta = 1 \implies \theta = 45^\circ\)).
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
\(\sin(90^\circ - \theta) = \cos \theta\).
Reason (R)
\(\tan(90^\circ - \theta) = \cot \theta\).
Correct Option: (B)
Both are true complementary angle identities.
Both are true complementary angle identities.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
The value of \(\tan 60^\circ\) is \(1/\sqrt{3}\).
Reason (R)
\(\tan 60^\circ = \sqrt{3}\) and \(\tan 30^\circ = 1/\sqrt{3}\).
Correct Option: (D)
Assertion (A) is false. Reason (R) is true.
Assertion (A) is false. Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The value of \(\sin 30^\circ + \cos 60^\circ\) is \(1\).
Reason (R)
\(\sin 30^\circ = 1/2\) and \(\cos 60^\circ = 1/3\).
Correct Option: (C)
Assertion (A) is true (\(1/2 + 1/2 = 1\)). Reason (R) is false because \(\cos 60^\circ = 1/2\), not \(1/3\).
Assertion (A) is true (\(1/2 + 1/2 = 1\)). Reason (R) is false because \(\cos 60^\circ = 1/2\), not \(1/3\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
\(\cos 0^\circ = 1\).
Reason (R)
\(\sin 0^\circ = 1\).
Correct Option: (C)
Assertion (A) is true (\(\cos 0^\circ = 1\)). Reason (R) is false because \(\sin 0^\circ = 0\), not \(1\).
Assertion (A) is true (\(\cos 0^\circ = 1\)). Reason (R) is false because \(\sin 0^\circ = 0\), not \(1\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
The value of \(\tan 0^\circ\) is \(0\).
Reason (R)
The value of \(\tan 90^\circ\) is undefined.
Correct Option: (B)
Both are true values for tangent function at boundary angles.
Both are true values for tangent function at boundary angles.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
If \(\sin A = \frac{1}{2}\), then \(A = 30^\circ\) (where \(A\) is an acute angle).
Reason (R)
\(\sin 30^\circ = \frac{1}{2}\).
Correct Option: (A)
Since \(A\) is acute and \(\sin 30^\circ = 1/2\), \(A = 30^\circ\). Reason explains Assertion.
Since \(A\) is acute and \(\sin 30^\circ = 1/2\), \(A = 30^\circ\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The value of \(\sec^2 \theta - \tan^2 \theta\) is always \(1\) for \(0^\circ \le \theta < 90^\circ\).
Reason (R)
By trigonometric identity \(1 + \tan^2 \theta = \sec^2 \theta\), transposing \(\tan^2 \theta\) gives \(\sec^2 \theta - \tan^2 \theta = 1\).
Correct Option: (A)
Identity \(1+\tan^2\theta = \sec^2\theta \implies \sec^2\theta - \tan^2\theta = 1\). Reason explains Assertion.
Identity \(1+\tan^2\theta = \sec^2\theta \implies \sec^2\theta - \tan^2\theta = 1\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
\(\sin(A + B) \ne \sin A + \sin B\) in general.
Reason (R)
\(\sin(30^\circ + 60^\circ) = \sin 30^\circ + \sin 60^\circ\).
Correct Option: (C)
Assertion (A) is true. Reason (R) is false because \(\sin(90^\circ) = 1\), whereas \(\sin 30^\circ + \sin 60^\circ = 1/2 + \sqrt{3}/2 = (1+\sqrt{3})/2 \ne 1\).
Assertion (A) is true. Reason (R) is false because \(\sin(90^\circ) = 1\), whereas \(\sin 30^\circ + \sin 60^\circ = 1/2 + \sqrt{3}/2 = (1+\sqrt{3})/2 \ne 1\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
If \(\tan \theta = \frac{4}{3}\), then \(\sin \theta = \frac{4}{5}\).
Reason (R)
If \(\tan \theta = \text{Opposite}/\text{Adjacent} = 4/3\), then Hypotenuse \(= \sqrt{4^2 + 3^2} = 5\), giving \(\sin \theta = \text{Opposite}/\text{Hypotenuse} = 4/5\).
Correct Option: (A)
Pythagorean triplet \((3, 4, 5)\) gives hypotenuse \(= 5\), so \(\sin\theta = 4/5\). Reason explains Assertion.
Pythagorean triplet \((3, 4, 5)\) gives hypotenuse \(= 5\), so \(\sin\theta = 4/5\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
\(\sin \theta\) increases from \(0\) to \(1\) as \(\theta\) increases from \(0^\circ\) to \(90^\circ\).
Reason (R)
\(\cos \theta\) decreases from \(1\) to \(0\) as \(\theta\) increases from \(0^\circ\) to \(90^\circ\).
Correct Option: (B)
Both are true monotonic behavior facts of sine and cosine in Quadrant I.
Both are true monotonic behavior facts of sine and cosine in Quadrant I.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
\(\operatorname{cosec}^2 \theta - \cot^2 \theta = 1\).
Reason (R)
\(\sin^2 \theta + \cos^2 \theta = 1\).
Correct Option: (B)
Both are true standard Pythagorean trigonometric identities.
Both are true standard Pythagorean trigonometric identities.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
If \(\sqrt{3}\tan \theta = 1\), then \(\theta = 30^\circ\).
Reason (R)
\(\tan 30^\circ = \sqrt{3}\).
Correct Option: (C)
Assertion (A) is true (\(\tan\theta = 1/\sqrt{3} \implies \theta = 30^\circ\)). Reason (R) is false because \(\tan 30^\circ = 1/\sqrt{3}\), whereas \(\tan 60^\circ = \sqrt{3}\).
Assertion (A) is true (\(\tan\theta = 1/\sqrt{3} \implies \theta = 30^\circ\)). Reason (R) is false because \(\tan 30^\circ = 1/\sqrt{3}\), whereas \(\tan 60^\circ = \sqrt{3}\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
\(\sin(A + B) = \sin A + \sin B\) for all angles \(A\) and \(B\).
Reason (R)
Trigonometric functions like sine are not linear operators; \(\sin(A+B) = \sin A \cos B + \cos A \sin B\).
Correct Option: (D)
Assertion (A) is false. Reason (R) is true.
Assertion (A) is false. Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The value of \(\tan 45^\circ + \cot 45^\circ = 2\).
Reason (R)
\(\tan 45^\circ = 1\) and \(\cot 45^\circ = 1\).
Correct Option: (B)
Both A and R are true (\(1 + 1 = 2\)).
Both A and R are true (\(1 + 1 = 2\)).
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
The maximum value of \(\sin \theta + \cos \theta\) is \(\sqrt{2}\).
Reason (R)
The maximum value of \(\sin \theta + \cos \theta\) occurs at \(\theta = 90^\circ\).
Correct Option: (C)
Assertion (A) is true (at \(\theta = 45^\circ\), \(\sin 45^\circ + \cos 45^\circ = 1/\sqrt{2} + 1/\sqrt{2} = \sqrt{2}\)). Reason (R) is false because at \(\theta = 90^\circ\), sum is \(1+0=1\), not maximum \(\sqrt{2}\).
Assertion (A) is true (at \(\theta = 45^\circ\), \(\sin 45^\circ + \cos 45^\circ = 1/\sqrt{2} + 1/\sqrt{2} = \sqrt{2}\)). Reason (R) is false because at \(\theta = 90^\circ\), sum is \(1+0=1\), not maximum \(\sqrt{2}\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
The value of \(\cos \theta\) increases as \(\theta\) increases from \(0^\circ\) to \(90^\circ\).
Reason (R)
\(\cos \theta\) decreases from \(1\) to \(0\) as \(\theta\) increases from \(0^\circ\) to \(90^\circ\).
Correct Option: (D)
Assertion (A) is false (it decreases). Reason (R) is true.
Assertion (A) is false (it decreases). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
\(\sqrt{1 - \cos^2 \theta} = \cos \theta\).
Reason (R)
By Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1 \implies 1 - \cos^2 \theta = \sin^2 \theta \implies \sqrt{1-\cos^2\theta} = \sin \theta\).
Correct Option: (D)
Assertion (A) is false (it equals \(\sin\theta\)). Reason (R) is true.
Assertion (A) is false (it equals \(\sin\theta\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
\(\sin^2 30^\circ + \cos^2 60^\circ = 1\).
Reason (R)
\(\sin^2 30^\circ + \cos^2 60^\circ = (1/2)^2 + (1/2)^2 = 1/4 + 1/4 = 1/2\).
Correct Option: (D)
Assertion (A) is false (it is \(1/2\), not \(1\)). Reason (R) is true.
Assertion (A) is false (it is \(1/2\), not \(1\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The value of \((1 + \tan^2 \theta)(1 - \sin^2 \theta) = 1\).
Reason (R)
\(1 + \tan^2 \theta = \cot^2 \theta\).
Correct Option: (C)
Assertion (A) is true (\(\sec^2\theta \times \cos^2\theta = 1\)). Reason (R) is false because \(1+\tan^2\theta = \sec^2\theta\), not \(\cot^2\theta\).
Assertion (A) is true (\(\sec^2\theta \times \cos^2\theta = 1\)). Reason (R) is false because \(1+\tan^2\theta = \sec^2\theta\), not \(\cot^2\theta\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
For any acute angle \(\theta\), \(\sin^2 \theta + \cos^2 \theta = 1\).
Reason (R)
In right-angled triangle \(\triangle ABC\) right-angled at \(B\), \(AB^2 + BC^2 = AC^2\). Dividing both sides by \(AC^2\) gives \(\left(\frac{AB}{AC}\right)^2 + \left(\frac{BC}{AC}\right)^2 = 1 \implies \sin^2 \theta + \cos^2 \theta = 1\).
Correct Option: (A)
The fundamental identity is proved via Pythagoras theorem in a right triangle. Reason explains Assertion.
The fundamental identity is proved via Pythagoras theorem in a right triangle. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If \(\sec \theta + \tan \theta = p\), then \(\sec \theta - \tan \theta = \frac{1}{p}\).
Reason (R)
Since \(\sec^2 \theta - \tan^2 \theta = 1 \implies (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = 1 \implies p(\sec \theta - \tan \theta) = 1\).
Correct Option: (A)
Difference of squares \((\sec\theta+\tan\theta)(\sec\theta-\tan\theta)=1\) proves \(\sec\theta-\tan\theta = 1/p\). Reason explains Assertion.
Difference of squares \((\sec\theta+\tan\theta)(\sec\theta-\tan\theta)=1\) proves \(\sec\theta-\tan\theta = 1/p\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
\(\sec A\) is the product of 'sec' and 'A'.
Reason (R)
'sec' is an abbreviation for secant and has no meaning independent of angle \(A\).
Correct Option: (D)
Assertion (A) is false. Reason (R) is true.
Assertion (A) is false. Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
If \(\sin A + \sin^2 A = 1\), then \(\cos^2 A + \cos^4 A = 1\).
Reason (R)
\(\sin^2 A + \cos^2 A = 2\).
Correct Option: (C)
Assertion (A) is true (\(\sin A = 1 - \sin^2 A = \cos^2 A \implies \cos^2 A + \cos^4 A = \sin A + \sin^2 A = 1\)). Reason (R) is false because \(\sin^2 A + \cos^2 A = 1\), not \(2\).
Assertion (A) is true (\(\sin A = 1 - \sin^2 A = \cos^2 A \implies \cos^2 A + \cos^4 A = \sin A + \sin^2 A = 1\)). Reason (R) is false because \(\sin^2 A + \cos^2 A = 1\), not \(2\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
If \(\cos \theta = \frac{1}{2}\), then \(\theta = 60^\circ\).
Reason (R)
If \(\tan \theta = 1\), then \(\theta = 45^\circ\).
Correct Option: (B)
Both statements are true specific angle values.
Both statements are true specific angle values.
Correct Answer: (B)
Both true.
Both true.