Mathematics Questions and Answers – Inverse Trigonometric Functions Basics

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Inverse Trigonometric Functions Basics”.

1. \(tan^{-1}\sqrt{3}+sec^{-1}⁡2 – cos^{-1}⁡1\) is equal to ________
a) 0
b) \(\frac{2π}{3}\)
c) \(\frac{π}{3}\)
d) \(\frac{π}{4}\)
View Answer

Answer: b
Explanation: \(tan^{-1}\sqrt{3}=\frac{π}{3}, sec^{-1}⁡2=\frac{π}{3}, cos^{-1}⁡1=0\)
∴\(tan^{-1}\sqrt{3}+sec^{-1}⁡2 -cos^{-1}⁡1=\frac{π}{3}+\frac{π}{3}-0\)
=\(\frac{2π}{3}\).
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2. What is the principle value of \(sec^{-1}⁡(\frac{2}{\sqrt{3}})\).
a) \(\frac{π}{6}\)
b) \(\frac{π}{3}\)
c) \(\frac{π}{4}\)
d) \(\frac{π}{2}\)
View Answer

Answer: a
Explanation: Let \(sec^{-1}⁡(\frac{2}{\sqrt{3}})\)=y
sec⁡ y=\(\frac{2}{\sqrt{3}}\)
sec⁡ y=sec \(⁡\frac{π}{6}\)
⇒y=\(\frac{π}{6}\)

3. What is the value of \(tan^1⁡\frac{1}{√3}-sin^{-1}⁡1+ cos^{-1}\frac{⁡1}{2}\) is ________
a) 2π
b) \(⁡\frac{π}{2}\)
c) π
d) 0
View Answer

Answer: c
Explanation: \(tan^{-1}⁡\frac{1}{\sqrt{3}}=\frac{π}{6},sin^{-1}⁡1=\frac{π}{2}, cos^{-1}\frac{⁡1}{2}=\frac{π}{3}\)
\(tan^1⁡\frac{1}{\sqrt{3}}-sin^{-1}⁡1+ cos^{-1}⁡\frac{1}{2}=\frac{π}{6}+\frac{π}{2}+\frac{π}{3}=\frac{π+3π+2π}{6}=\frac{6π}{6}=π\)
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4. [-1, 1] is the domain for which of the following inverse trigonometric functions?
a) sin-1⁡x
b) cot-1⁡x
c) tan-1⁡x
d) sec-1⁡x
View Answer

Answer: a
Explanation: [-1, 1] is the domain for sin-1⁡x.
The domain for cot-1⁡x is (-∞,∞).
The domain for tan-1⁡x is (-∞,∞).
The domain for sec-1⁡x is (-∞,-1]∪[1,∞).

5. The domain of sin-1⁡(3x) is equal to _______
a) [-1, 1]
b) \([\frac{-1}{3}, \frac{1}{3}]\)
c) [-3, 3]
d) [-3π, 3π]
View Answer

Answer: b
Explanation: The domain of y=sin-1⁡x is -1≤x≤1.
∴the domain of y=sin-1⁡3x is-1≤3x≤1
⇒ \([\frac{-1}{3} ≤ x ≤ \frac{1}{3}]\)
Hence, \([\frac{-1}{3}, \frac{1}{3}]\).
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6. What is the value of 5 \(cos^{-1}\frac{⁡1}{2} + 7 sin^{-1}⁡(\frac{-1}{2})\) ?
a) –\(\frac{π}{2}\)
b) π
c) \(\frac{π}{2}\)
d) \(\frac{17π}{6}\)
View Answer

Answer: c
Explanation: \(cos^{-1}⁡(\frac{⁡1}{2})=\frac{π}{3} and sin^{-1}⁡(-\frac{⁡1}{2})=-\frac{π}{6}\)
∴ 5 \(cos^{-1}⁡\frac{1}{2}+7 sin^{-1}⁡(-\frac{1}{2}) =5(\frac{π}{3})+7(-\frac{π}{6})\)
=\(\frac{5π}{3}-\frac{7π}{6}=\frac{10π-7π}{6}=\frac{3π}{6}=\frac{π}{2}\)

7. Find the value of \(sin^{-1}⁡(sin⁡ \frac{4π}{3})\) is _______
a) π
b) \(\frac{π}{3}\)
c) \(\frac{4π}{3}\)
d) –\(\frac{π}{3}\)
View Answer

Answer: d
Explanation: \(sin^{-1}⁡(sin⁡x)\)=x, x∈\([-\frac{π}{2},\frac{π}{2}]\)
∴\(sin^{-1} (sin⁡ \frac{4π}{3})=sin^{-1}⁡(sin⁡(π+\frac{π}{3}))=sin^{-1}⁡(sin⁡(\frac{-π}{3}))= -\frac{π}{3}\).
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8. Find the value of \(cos⁡(sin^{-1}⁡\frac{\sqrt{3}}{2})\) is _____
a) \(\frac{\sqrt{3}}{2}\)
b) \(\frac{1}{4}\)
c) \(\frac{1}{2}\)
d) 0
View Answer

Answer: c
Explanation: \(sin⁡\frac{π}{3}=\frac{\sqrt{3}}{2}\)
∴\(sin^{-1}⁡\frac{\sqrt{3}}{2}=\frac{π}{3}\)
⇒\(cos⁡(sin^{-1}⁡\frac{\sqrt{3}}{2})=cos⁡(\frac{π}{3})=\frac{1}{2}\).

9. If \(cos^{-1}⁡x=y\), then which of the following is correct?
a) 0 ≤ y ≤ π
b) 0 < y < π
c) –\(\frac{π}{2}≤y≤\frac{π}{2}\)
d) –\(\frac{π}{2}<y<\frac{π}{2}\)
View Answer

Answer: a
Explanation: Given that, \(cos^{-1}⁡x=y\)
The range of principle values for the inverse trigonometric function \(cos^{-1}\) is [0,π].
Hence, 0≤y≤π.
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10. \(sin^{-1}⁡x\) is same as \((sin⁡x)^{-1}\).
a) True
b) False
View Answer

Answer: b
Explanation: The given statement is false. \(sin^{-1}⁡x\) is not same as \((sin⁡x)^{-1}\). \(sin^{-1}⁡x\) is an inverse trigonometric function whereas \((sin⁡x)^{-1}\) is just the reciprocal of sin⁡x i.e. \(sin⁡x=\frac{1}{sin⁡x}\).

Sanfoundry Global Education & Learning Series – Mathematics – Class 12.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter