Solution of Trigonometric Equations
A method for finding angles.
(i) First of all, we determine the quadrant where the angle falls. For this, we use the all sin, tan, cos rule.
If sinθ is positive, the angleθ falls in the 1st and 2nd quadrants and if sinθ is negative, the angleθ falls in the 3rd and 4th quadrants. If cosθ is positive,θ lies in the 1st and 4th quadrants and if cosθ is negative,θ lies in the 2nd and 3rd quadrants. if tanθ is positive,θ lies in the first and third quadrants and if tanθ is negative,θ lies in second and fourth quadrants.
(ii) To find the angle in the first quadrant, we find the acute angle which satisfies the equation.
For example, if 2cosθ = 1
then cosθ = \(\frac{1}{2}\)
or, cosθ = cos600
So, θ = 600
(iii) To find the angle in the second quadrant, we subtract acute angleθ from 1800.
(iv) To find the angle in the third quadrant, we add acute angleθ to 1800
(v) To find the angle in the 4th quadrant, we subtract acute angleθ to 3600
(vi) To find the value ofθ from the equations like sinθ = 0, cosθ = 0, tanθ= 0, sinθ = 1, cosθ = 1, sinθ = -1, cosθ = -1. We should note the following results:
| If sinθ = 0, thenθ = 00, 1800 or 3600 | If sinθ = 1, thenθ = 900 |
| If tanθ = 0, thenθ = 00,1800 or 3600 | If sinθ = -1, thenθ = 2700 |
| If cosθ = 0, thenθ = 900 or 2700 | If cosθ = 1, thenθ = 00 or 3600 |
If cosθ = -1, then θ = 1800.
Summary
A method for finding angles.
(i) First of all, we determine the quadrant where the angle falls. For this, we use the all sin, tan, cos rule.
If sinθ is positive, the angleθ falls in the 1st and 2nd quadrants and if sinθ is negative, the angleθ falls in the 3rd and 4th quadrants. If cosθ is positive,θ lies in the 1st and 4th quadrants and if cosθ is negative,θ lies in the 2nd and 3rd quadrants. if tanθ is positive,θ lies in the first and third quadrants and if tanθ is negative,θ lies in second and fourth quadrants.
(ii) To find the angle in the first quadrant, we find the acute angle which satisfies the equation.
For example, if 2cosθ = 1
then cosθ = \(\frac{1}{2}\)
or, cosθ = cos600
So, θ = 600
(iii) To find the angle in the second quadrant, we subtract acute angleθ from 1800.
(iv) To find the angle in the third quadrant, we add acute angleθ to 1800
(v) To find the angle in the 4th quadrant, we subtract acute angleθ to 3600
(vi) To find the value ofθ from the equations like sinθ = 0, cosθ = 0, tanθ= 0, sinθ = 1, cosθ = 1, sinθ = -1, cosθ = -1. We should note the following results:
| If sinθ = 0, thenθ = 00, 1800 or 3600 | If sinθ = 1, thenθ = 900 |
| If tanθ = 0, thenθ = 00,1800 or 3600 | If sinθ = -1, thenθ = 2700 |
| If cosθ = 0, thenθ = 900 or 2700 | If cosθ = 1, thenθ = 00 or 3600 |
If cosθ = -1, then θ = 1800.
Things to Remember
A method for finding angles
(i) First of all, we determine the quadrant where the angle falls. For this, we use the all sin, tan, cos rule.
(ii) To find the angle in the first quadrant, we find the acute angle which satisfies the equation.
(iii) To find the angle in the second quadrant, we subtract acute angle θ from 1800.
(iv) To find the angle in the third quadrant, we add acute angle θ to 1800
(v) To find the angle in the fourth quadrant, we subtract acute angle θ to 3600
MCQs
No MCQs found.
Subjective Questions
Q1:
What is a profession?
Type: Very_short Difficulty: Easy
Q2:
What are the types of profession on the basis of nature?
Type: Very_short Difficulty: Easy
<ul>
<li>Generalist and Specialist</li>
<li>Production-oriented and Service-oriented</li>
</ul>
Q3:
What is a training?
Type: Very_short Difficulty: Easy
Q4:
What is an orientation training?
Type: Very_short Difficulty: Easy
Q5:
What is a safety training?
Type: Very_short Difficulty: Easy
Q6:
What is a promotional training?
Type: Very_short Difficulty: Easy
Q7:
What is a refresher's training?
Type: Very_short Difficulty: Easy
Q8:
What is a remedial training?
Type: Very_short Difficulty: Easy
Q9:
What is an internship training?
Type: Very_short Difficulty: Easy
Q10:
Explain the types of training.
Type: Long Difficulty: Easy
<ul>
<li><strong>Orientation Training</strong><br />It is a basic type of training which is usually of very short period i.e. few hours or few days. It gives a general idea about any specific subject or an organization.</li>
<li><strong>On the Job Training</strong><br />It is the training which is provided to an employee during his/her job tenure. It helps to develop skills and knowledge required to build up a capacity of the employee to perform his/her job and to carry out his responsibilities.</li>
<li><strong>Safety Training</strong><br />It is a type of training which is provided in order to develop skills to reduce accidents, to protect the health and to be safe from possible accidents.</li>
<li><strong>Promotional Training</strong><br />It is a type of training which is provided to an employee when he/she is promoted to senior position or as a prerequisite for getting a promotion.</li>
<li><strong>Refreshers Training</strong><br />It is a training which is provided to an individual in the field in which he/she is well known. This type of training is provided to an employee in order to maintain his/her efficiency.</li>
<li><strong>Remedial Training</strong><br />It is a type of training which is provided when any drawback or dissatisfaction is observed in employees' behavior. It is provided in order to improve the behavior of an individual.</li>
<li><strong>Internship Training</strong><br />It is a type of training which is provided immediately to an individual after his/her completion of formal professional education and before joining a professional job. This training is of longer time in comparison to other types of training.</li>
</ul>
<p> </p>
Q11:
Explain the types of profession.
Type: Short Difficulty: Easy
<ul>
<li><strong>Generalist and Specialist<br /></strong>Generalists are the persons with a broad general knowledge. They have superficial knowledge in several areas and also have the ability to combine ideas from various fields. For example, administrative posts like section officers, Nayab Subba etc.<br />Specialists are the persons who are an expert in, or dedicated to, some specific branch of study or research. They have narrow coverage but a very large depth of knowledge and skill. For example, technical posts like vet doctors, agronomists (a scientist whose speciality id agronomy), civil engineers etc.</li>
<li><strong>Production-oriented and service-oriented<br /></strong>Professions are also production-oriented and service-oriented as business activities. People are engaged in making certain products or goods in the production-oriented profession whereas people use their skills and knowledge for serving the people and the society in the service-oriented profession. For example, vegetable farming, knitting, weaving, etc. are production-oriented professions whereas, tourism service, telecommunication service, civil service, etc. are the service-oriented professions.</li>
</ul>
Q12:
Why is training important for professionals?
Type: Short Difficulty: Easy
Videos
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Training and Development
Solution of Trigonometric Equations
Equalities like 1 - sin2θ = cos2θ , 1 + cos2θ = 2 cos2θ etc are satisfied by all the values of the angle θ . So, they are called identities.
Let the equalities like 2cosθ = 1,\(\sqrt 2\) sinθ = 1 etc. are satisfied by only some values of the angle θ.
A trigonometric ratio of a certain angle has one and only value. But, if the value of a trigonometric ratio is given, the angle is not unique. For example, let us take the equation
2cosx = \(\sqrt 3\) or cos x = \(\frac{\sqrt 3}{2}\)
We know that cos300 = \(\frac{\sqrt 3}{2}\)
So, x = 300
Again, cos3300= cos(3600 - 300) = cos300 = \(\frac{\sqrt 3}{2}\)
So, x = 3300
∴ x may be 300 or 3300.
Again if we add 3600 or multiple of 3600 to the angle 300 or 3300, the cosine of any one of these angles will also be \(\frac{\sqrt 3}{2}\). So there might be many values of x which satisfy the equation. But we will try to find those angles which lie between 00 and 3600.
A method for finding angles:
(i) First of all, we determine the quadrant where the angle falls. For this, we use the all sin, tan, cos rule.
If sinθ is positive, the angle θ falls in the 1st and 2nd quadrants and if sinθ is negative, the angle θ falls in the 3rd and 4th quadrants. If cosθ is positive, θ lies in the 1st and 4th quadrants and if cosθ is negative, θ lies in the 2nd and 3rd quadrants. If tanθ is positive, θ lies in the first and third quadrants and if tanθ is negative, θ lies in second and fourth quadrants.
(ii) To find the angle in the first quadrant, we find the acute angle which satisfies the equation.
For example, if 2cosθ = 1
then cosθ = \(\frac{1}{2}\)
or, cosθ = cos600
So, θ = 600
(iii) To find the angle in the second quadrant, we subtract acute angle θ from 1800.
(iv) To find the angle in the third quadrant, we add acute angle θ to 1800
(v) To find the angle in the fourth quadrant, we subtract acute angle θ to 3600
(vi) To find the value of θ from the equations like sinθ = 0, cosθ = 0, tanθ= 0, sinθ = 1, cosθ = 1, sinθ = -1, cosθ = -1. We should note the following results:
| If sinθ = 0, thenθ = 00, 1800 or 3600 | If sinθ = 1, thenθ = 900 |
| If tanθ = 0, thenθ = 00,1800 or 3600 | If sinθ = -1, thenθ = 2700 |
| If cosθ = 0, thenθ = 900 or 2700 | If cosθ = 1, thenθ = 00 or 3600 |
| If cosθ = -1, then θ = 1800. |
Lesson
Trigonometry
Subject
Optional Mathematics
Grade
Grade 10
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