Evaluate : ^-1 ((3π)/(4) ) + ^-1 ((3π)/(4) ) + ^-1(1)

MathsCBSE Class 12 · Board exam 2023, set 65/1/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2023: question 21(b)

Set in the CBSE Class 12 board exam of 2023, paper 65/1/1, section B, question 21(b), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q21(b) · 2 marks · Very short answerInverse Trigonometric Functions

Evaluate : $sin^{- 1} (sin \frac{3 π}{4}) + cos^{- 1} (cos \frac{3 π}{4}) + tan^{- 1} (1)$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 65/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Recall the principal value ranges: $[- π /2, π /2]$ for $sin^{- 1}$ and $[0, π]$ for $cos^{- 1}$ .

Hint 2

$3 π /4$ is not in the range of $sin^{- 1}$ . Find an angle in that range with the same sine.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

$\frac{3 π}{4} \in / [- \frac{π}{2}, \frac{π}{2}]$ . Since $sin \frac{3 π}{4} = sin \frac{π}{4}$ , we get $sin^{- 1} (sin \frac{3 π}{4}) = \frac{π}{4}$ .
$\frac{3 π}{4} \in [0, π]$ , so $cos^{- 1} (cos \frac{3 π}{4}) = \frac{3 π}{4}$ .
Also $tan^{- 1} (1) = \frac{π}{4}$ .
Required value $= \frac{π}{4} + \frac{3 π}{4} + \frac{π}{4}$ .
1½ marks
$= \frac{5 π}{4}$
½ mark

Answer: $\frac{5 π}{4}$

Where the marks are

Writes the three principal values $\frac{π}{4}, \frac{3 π}{4}, \frac{π}{4}$ (1½ marks)
Final sum $\frac{5 π}{4}$ (½ mark)

Revise the chapter

Chapter: Inverse Trigonometric Functions (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Inverse Trigonometric Functions, one at a time, marked against the answer key and marking points.Back to the full paper2023 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2023: question 21(b)

Hints

Step-by-step solution

Revise the chapter

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