SaraNextGen.Com
Updated By SaraNextGen
On January 25, 2024, 11:35 AM

Exercise 2.2 - Chapter 2 Inverse Trigonometry class 12 ncert solutions Maths - SaraNextGen [2024]


Question 1:

Prove https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8236/Chapter%202_html_m5f173286.gif

Answer:

To prove: https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8236/Chapter%202_html_m5f173286.gif

Let x = sinθ. Then, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8236/Chapter%202_html_43d4773b.gif

We have,

R.H.S. =https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8236/Chapter%202_html_med71555.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8236/Chapter%202_html_m12ab92b3.gif

= 3θ

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8236/Chapter%202_html_m4406dcf8.gif

= L.H.S.

Question 2:

Prove https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8237/Chapter%202_html_m1e80cc21.gif

Answer:

To prove:https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8237/Chapter%202_html_m1e80cc21.gif

Let x = cosθ. Then, cos−1 x =θ.

We have,

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8237/Chapter%202_html_6f7056ef.gif

Question 3:

Prove https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8238/Chapter%202_html_m7961d769.gif

Answer:

To prove:https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8238/Chapter%202_html_m7961d769.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8238/Chapter%202_html_m2b8918f0.gif

Question 4:

Prove https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8239/Chapter%202_html_3c045b5e.gif

Answer:

To prove: https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8239/Chapter%202_html_3c045b5e.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8239/Chapter%202_html_261ec9f5.gif

Question 5:

Write the function in the simplest form:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8240/Chapter%202_html_13d7e339.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8240/Chapter%202_html_7c792e6d.gif

Question 6:

Write the function in the simplest form:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8241/Chapter%202_html_44000ff6.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8241/Chapter%202_html_44000ff6.gif

Put x = cosec θ ⇒ θ = cosec−1 x

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8241/Chapter%202_html_76ec630a.gif

Question 7:

Write the function in the simplest form:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8242/Chapter%202_html_44231546.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8242/Chapter%202_html_4c718ec7.gif

Question 8:

Write the function in the simplest form:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8243/Chapter%202_html_m4c6bdcfa.gif

Answer:

tan-1cosx-sinxcosx+sinx=tan-11-sinxcosx1+sinxcosx=tan-11-tanx1+tanx=tan-11-tan-1tanx    tan-1x-y1+xy=tan-1x-tan-1y=π4-x

Question 9:

Write the function in the simplest form:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8244/Chapter%202_html_m87c9839.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8244/Chapter%202_html_m6e536503.gif

Question 10:

Write the function in the simplest form:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8245/Chapter%202_html_494cb76b.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8245/Chapter%202_html_7e7fb12f.gif

Question 11:

Find the value of https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8246/Chapter%202_html_64df0e7a.gif

Answer:

Lethttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8246/Chapter%202_html_m26c9b49.gif. Then,https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8246/Chapter%202_html_mefaf402.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8246/Chapter%202_html_10923d49.gif

Question 12:

Find the value of https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8247/Chapter%202_html_m1a481954.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8247/Chapter%202_html_52443149.gif

Question 13:

Find the value of https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8248/Chapter%202_html_m20f9e382.gif

Answer:

Let x = tan θ. Then, θ = tan−1 x.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8248/Chapter%202_html_c7f5408.gif

Let y = tan Φ. Then, Φ = tan−1 y.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8248/Chapter%202_html_245cd23b.gif

Question 14:

Ifhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8249/Chapter%202_html_4a5af6ac.gif, then find the value of x.

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8249/Chapter%202_html_m2360a54e.gif

On squaring both sides, we get:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8249/Chapter%202_html_mdbee6ce.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8249/Chapter%202_html_6fbad75c.gif

Hence, the value of x ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8249/Chapter%202_html_m1f05a0cd.gif

Question 15:

Ifhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8250/Chapter%202_html_m1a20e5ca.gif, then find the value of x.

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8250/Chapter%202_html_m36820ccf.gif

Hence, the value of x is https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8250/Chapter%202_html_m76605374.gif

Question 16:

Find the values of https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8251/Chapter%202_html_m1df80c55.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8251/Chapter%202_html_m1df80c55.gif

We know that sin−1 (sin x) = x ifhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8251/Chapter%202_html_426a3638.gif, which is the principal value branch of sin−1x.

Here,https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8251/Chapter%202_html_m1ed0bee4.gif

Now, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8251/Chapter%202_html_m1df80c55.gifcan be written as:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8251/Chapter%202_html_m56e77f67.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8251/Chapter%202_html_m459728e3.gif

Question 17:

Find the values of https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8252/Chapter%202_html_m496170f3.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8252/Chapter%202_html_m496170f3.gif

We know that tan−1 (tan x) = x ifhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8252/Chapter%202_html_m448424b2.gif, which is the principal value branch of tan−1x.

Here,https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8252/Chapter%202_html_m75a4fa3b.gif

Now, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8252/Chapter%202_html_m496170f3.gifcan be written as:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8252/Chapter%202_html_75479362.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8252/Chapter%202_html_155e1c32.gif

Question 18:

Find the values of https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8253/Chapter%202_html_626d4d8e.gif

Answer:

Lethttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8253/Chapter%202_html_7835c198.gif. Then,https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8253/Chapter%202_html_m6b228fa3.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8253/Chapter%202_html_2b0e4f31.gif

Question 19:

Find the values of https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8254/Chapter%202_html_274387a8.gifis equal to

(A) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8254/Chapter%202_html_31b827a5.gif (B) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8254/Chapter%202_html_m4e248d0.gif (C) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8254/Chapter%202_html_m220be517.gif (D) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8254/Chapter%202_html_5bc0b8c.gif

Answer:

We know that cos−1 (cos x) = x ifhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8254/Chapter%202_html_1db67982.gif, which is the principal value branch of cos −1x.

Here,https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8254/Chapter%202_html_345d6c9.gif

Now, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8254/Chapter%202_html_274387a8.gifcan be written as:

 

cos-1cos7π6 = cos-1cosπ+π6cos-1cos7π6 = cos-1- cosπ6  as, cosπ+θ = – cos θcos-1cos7π6  = cos-1- cosπ-5π6cos-1cos7π6 = cos-1– cos 5π6   as, cosπ-θ = – cos θhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8254/Chapter%202_html_31f7ea15.gif

The correct Answer is B.

Question 20:

Find the values of https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_607ccadd.gifis equal to

(A) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_eeecab0.gif (B) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_33f00ded.gif (C) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_682bd651.gif (D) 1

Answer:

Lethttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_m2bc5b727.gif. Then, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_m7ab2232f.gif

We know that the range of the principal value branch ofhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_2d4d89cf.gif.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_m25255b6c.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8255/Chapter%202_html_68b41bac.gif

The correct Answer is D.

Question 21:

Find the values of https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_m791a90d7.gifis equal to

(A) π (B) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_m4d28d5df.gif (C) 0 (D) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_m465f17c8.gif

Answer:

Lethttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_7f97e1a5.gif. Then,https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_27471169.gif

We know that the range of the principal value branch ofhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_1ad054a8.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_622d697f.gif

Lethttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_m631d3ca8.gif.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_m514fe183.gif

The range of the principal value branch ofhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_53b51094.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8256/Chapter%202_html_754fcfa8.gif

The correct Answer is B.

Also Read : Miscellaneous-Exercise-Chapter-2-Inverse-Trigonometry-class-12-ncert-solutions-Maths

SaraNextGen