
If\\[\cos \theta = - \frac{1}{{\sqrt 2 }}\] \and\\[\tan \theta = 1\], then the general value of\[\theta \]\\is
A. \[2n\pi + \frac{\pi }{4}\]
В. \[(2n + 1)\pi + \frac{\pi }{4}\]
C. \[n\pi + \frac{\pi }{4}\]
D. \[n\pi \pm \frac{\pi }{4}\]
Answer
164.4k+ views
Hint: Sometimes anything will become a trig identity by factoring with a widely used phrase. Multiply the conjugate by the denominator. Find the least common factor. A fraction can be divided into two smaller fractions. Everything should be rewritten using sine and cosine. Continue by utilizing trigonometric identities to solve the equation. In this case, \[\cos \theta = - \frac{1}{{\sqrt 2 }}\]and\[\tan \theta = 1\], first step is that you can factor. In order to recall this, we must be aware of the fundamental coordinate system used by all trigonometric functions as well as their general solution formula.
Complete step by step solution:We have given the equation according to the question:
\[\cos \theta = - \frac{1}{{\sqrt 2 }}\]and\\[\tan \theta = 1\]
Rewrite the equation given in terms of\[cos\]:
\[\cos \theta = - 1/\sqrt 2 = - \cos (\pi /4)\]
Solve to obtain the required values in order to find the value of theta:
\[ = \cos (\pi - \pi /4){\rm{ or }}\cos (\pi + \pi /4)\]
Determine the value for\[\theta \]:
\[\therefore \theta = 3\pi /4,5\pi /4\]
Now solve the equation in terms of tan:
\[\tan \theta = 1 = \tan (\pi /4),\tan (\pi + \pi /4)\]
Determine the value for\[\theta \]:
\[\therefore \theta = \pi /4,5\pi /4\]
Hence the value of\[\theta \]between\[0\]and \[2\pi \]which satisfies both the equations is\[5\pi /4\]
\[\therefore \]General value of the given equation is\[2n\pi + 5\pi /4 = (2{\rm{n}} + 1)\pi + \pi /4\].
Hence, the general value of\[\theta \]is\[(2n + 1)\pi + \frac{\pi }{4}\]
Option ‘B’ is correct
Note: Student makes errors while applying identities. Not able to comprehend how to arrive at the other side Students at times apply identities incorrectly \[sin\theta = 1 - cos\theta \]or\[tan\theta = 1 + {\rm{ }}sec\theta \]. Start the simplification of the equation, on both the sides simultaneously.
Due to the fact that this value is repeated after every n interval, we can only choose one common value of theta. And utilizing graphical techniques, we can discover this.
Complete step by step solution:We have given the equation according to the question:
\[\cos \theta = - \frac{1}{{\sqrt 2 }}\]and\\[\tan \theta = 1\]
Rewrite the equation given in terms of\[cos\]:
\[\cos \theta = - 1/\sqrt 2 = - \cos (\pi /4)\]
Solve to obtain the required values in order to find the value of theta:
\[ = \cos (\pi - \pi /4){\rm{ or }}\cos (\pi + \pi /4)\]
Determine the value for\[\theta \]:
\[\therefore \theta = 3\pi /4,5\pi /4\]
Now solve the equation in terms of tan:
\[\tan \theta = 1 = \tan (\pi /4),\tan (\pi + \pi /4)\]
Determine the value for\[\theta \]:
\[\therefore \theta = \pi /4,5\pi /4\]
Hence the value of\[\theta \]between\[0\]and \[2\pi \]which satisfies both the equations is\[5\pi /4\]
\[\therefore \]General value of the given equation is\[2n\pi + 5\pi /4 = (2{\rm{n}} + 1)\pi + \pi /4\].
Hence, the general value of\[\theta \]is\[(2n + 1)\pi + \frac{\pi }{4}\]
Option ‘B’ is correct
Note: Student makes errors while applying identities. Not able to comprehend how to arrive at the other side Students at times apply identities incorrectly \[sin\theta = 1 - cos\theta \]or\[tan\theta = 1 + {\rm{ }}sec\theta \]. Start the simplification of the equation, on both the sides simultaneously.
Due to the fact that this value is repeated after every n interval, we can only choose one common value of theta. And utilizing graphical techniques, we can discover this.
Recently Updated Pages
Environmental Chemistry Chapter for JEE Main Chemistry

Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Main Physics Kinematics for JEE Main 2025

Difference Between Natural and Whole Numbers: JEE Main 2024

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

Degree of Dissociation and Its Formula With Solved Example for JEE

Instantaneous Velocity - Formula based Examples for JEE

JEE Main 2025: Conversion of Galvanometer Into Ammeter And Voltmeter in Physics

JEE Advanced 2025 Notes
