Like us!

Have a Look!

WhatsApp Message

Share Link

Trigonometric ratios for 45 degrees

Introduction to Trigonometry I - Concepts

Identifying Hypotenuses In A Right Triangle

Identifying Adjacent Side To An Angle

Identifying Opposite Side To An Angle

Trigonometric Ratios

Reciprocal Trigonometric Ratios

Trigonometric ratios for 30 degrees

Trigonometric ratios for 45 degrees

Trigonometric ratios for 60 degrees

Trigonometric ratios for 90 degrees

Simplifying Trigonometric Expressions

Class - RRB Technician Grade I Signal Subjects

Concept Explanation

Trigonometric ratios for 45 degrees

Trigonometric Ratios for 45 Degrees: Value for the Trigonometrical Ratios for certain angle such as $30^0$ , $60^0$ , $90^0$ and $45^0$ are commonly called standard angles and the trigonometrical ratios of these angles are frequently used to solve particular angles. In this section we will derive the value of trigonometric ratios for $45^0$ .

Let $Delta ABC$ be a right - angled at B and let angle A is $45^0$

Now $angle B=90^o ; and ; angle A=45^o.$ Then, clearly, $angle C=45^o.$ [ Angle Sum Property]

$angle A=angle C ; Rightarrow ;AB=BC$ [sides opposite equal angles]

Let AB = BC = a units.

By the Pythagorean theoram

$AC=sqrt {AB^2+BC^2}; ; units ;; =sqrt {a^2+a^2}$

$AC=sqrt {2a^2} = sqrt 2; a$

Thus, Adjacent = AB = a units, Opposite = BC = a units and hypotenuse AC = $sqrt 2;$ a units.

Thus, we have

$sin ; 45^o =frac{Opposite}{Hypotenuse}=frac {BC}{AC}= frac {a}{sqrt {2}a}=frac {1}{sqrt {2}}$

$cos ; 45^o =frac{Adjacent}{Hypoteneuse}=frac {AB}{AC}=frac {a}{sqrt {2}a}=frac {1}{sqrt {2}}$

$tan ; 45^o =frac{Opposite}{Adjacent}=frac {BC}{AB}=frac {a}{a}=1$

$cosec ; 45^o=frac {1}{sin ; 45^o}=sqrt {2}$

$sec ; 45^o =frac {1}{cos; 45^o}=sqrt {2}$

$cot ; 45^o=frac {1}{tan ; 45^o}=1$

$large angle A$	sin A	cos A	tan A	cosec A	sec A	cot A
$large 45^{circ}$	$large frac{1}{sqrt{2}}$	$large frac{1}{sqrt{2}}$	1	$large {sqrt{2}}$	$large {sqrt{2}}$	$large 1$

Illustration: Simplify the given expression $sin ;45^0 + cos ;45^0;(1+ sin; 45^0;cos;45^0)$

Solution: To solve this expression we will substitute the value of the ratios at $45^0$

$sin ;45^0 + cos ;45^0;(1+ sin; 45^0;cos;45^0)$

$=frac{1}{sqrt 2}+frac{1}{sqrt 2}(1+frac{1}{sqrt 2}times frac{1}{sqrt2})$

$=frac{1}{sqrt 2}+frac{1}{sqrt 2}(1+frac{1}{2})$

$=frac{1}{sqrt 2}+frac{1}{sqrt 2}times frac{3}{2}$

$=frac{1}{sqrt 2}times (1+ frac{3}{2})$

$=frac{1}{sqrt 2}times frac{5}{2}$

$=frac{5}{2sqrt 2}$

Sample Questions

(More Questions for each concept available in Login)

Question : 1

$frac {1-tan^245^o}{1+tan^245^o}is : : equal : : to$
A. $tan : 90^o$	B. $1$	C. $sin: : 45^o$	D. $sin: : 0^o$
Right Option : D
View Explanation

Question : 2

Solve the following expressions : $(i);2(Cos^245^0+Tan^245^0)+5Cot^245^0$ $(ii);2(Cos^445^0+Sin^445^0)+3Sec^245^0$
A. 9 , 7	B. 6 , 9	C. 2 , 4	D. 4 , 2
Right Option : A
View Explanation

Question : 3

Solve the given expression : $sin;45^0+cos;45^0+(2+tan ;45^0+cot;45^0)$
A. $frac{5sqrt2+8}{2}$	B. $frac{2sqrt2+5}{2}$	C. $frac{2sqrt3+8}{3}$	D. $frac{2sqrt2+8}{2}$
Right Option : D
View Explanation

Students / Parents Reviews [20]

My experience with Abhyas academy is very good. I did not think that my every subject coming here will be so strong. The main thing is that the online tests had made me learn here more things.

Hiya Gupta

8th

One of the best institutes to develope a child interest in studies.Provides SST and English knowledge also unlike other institutes. Teachers are co operative and friendly online tests andPPT develo...

Aman Kumar Shrivastava

10th

Abhyas is an institute of high repute. Yogansh has taken admission last year. It creates abilities in child to prepare for competitive exams. Students are motivated by living prizes on basis of per...

Yogansh Nyasi

7th

It was a good experience with Abhyas Academy. I even faced problems in starting but slowly and steadily overcomed. Especially reasoning classes helped me a lot.

Cheshta

10th

My experience was very good with Abhyas academy. I am studying here from 6th class and I am satisfied by its results in my life. I improved a lot here ahead of school syllabus.

Ayan Ghosh

8th

Being a parent, I saw my daughter improvement in her studies by seeing a good result in all day to day compititive exam TMO, NSO, IEO etc and as well as studies. I have got a fruitful result from m...

Prisha Gupta

8th

Abhyas is a complete education Institute. Here extreme care is taken by teacher with the help of regular exam. Extra classes also conducted by the institute, if the student is weak.

Om Umang

10th

Abhyas institute is one of the best coaching institute in the vicinity of Ambala Cantt area. The teachers of the institute are well experienced and very helpful in solving the problems of the students.The good thing of the institute is that it is providing extra classes for the students who are w...

Aman Kumar Shrivastava

10th

It was good as the experience because as we had come here we had been improved in a such envirnment created here.Extra is taught which is beneficial for future.

Eshan Arora

8th

My experience with Abhyas is very good. I have learnt many things here like vedic maths and reasoning also. Teachers here first take our doubts and then there are assignments to verify our weak poi...

Shivam Rana

7th

The experience was nice. I studied here for three years and saw a tremendous change in myself. I started liking subjects like English and SST which earlier I ran from. Extra knowledge gave me confi...

Aman Kumar Shrivastava

10th

We started with lot of hope that Abhyas will help in better understnding of complex topics of highers classes. we are not disappointed with the progress our child has made after attending Abhyas. Though need to mention that we expected a lot more. On a scale of 1-10, we would give may be 7.

Manya

8th

Abhyas academy is great place to learn. I have learnt a lot here they have finished my fear of not answering.It has created a habit of self studying in me.The teachers here are very supportive and helpful. Earlier my maths and science was good but now it has been much better than before.

Barkha Arora

10th

Abhyas is good institution and a innovative institute also. It is a good platform of beginners.Due to Abhyas,he has got knoweledge about reasoning and confidence.My son has improved his vocabulary ...

Manish Kumar

10th

Abhyas institute is one of the best coaching institute in the vicinity of Ambala cantt.The institute provides good and quality education to the students.The teachers are well experienced and are v...

Abhyas institute is one of the best coaching institute in the vicinity of Ambala cantt.The institute provides good and quality education to the students.The teachers are well experienced and are very helpful in solving the problems. The major advantages of the institute is extra classes for weak...

Shreya Shrivastava

8th

When I have not joined Abhyas Academy, my skills of solving maths problems were not clear. But, after joining it, my skills have been developed and my concepts of science and SST are very well. I a...

Sharandeep Singh

7th

A marvelous experience with Abhyas. I am glad to share that my ward has achieved more than enough at the Ambala ABHYAS centre. Years have passed on and more and more he has gained. May the centre f...

Archit Segal

7th

In terms of methodology I want to say that institute provides expert guidence and results oriented monitering supplements by requsite study material along with regular tests which help the students to improve their education skills.The techniques of providing education helps the students to asses...

Aman Kumar Shrivastava

10th

My experience with Abhyas academy is very nice or it can be said wonderful. I have been studying here from seven class. I have been completing my journey of three years. I am tinking that I should ...

Hridey Preet

9th

Third consective year,my ward is in Abhyas with nice experience of admin and transport support.Educational standard of the institute recumbent at satisfactory level. One thing would live to bring in notice that last year study books was distributed after half of the session was over,though study ...

Ayan Ghosh

8th

View All

Attention !!!!!

Trigonometric ratios for 45 degrees

Trigonometric ratios for 45 degrees

Students / Parents Reviews [20]

Hiya Gupta

Aman Kumar Shrivastava

Yogansh Nyasi

Cheshta

Ayan Ghosh

Prisha Gupta

Om Umang

Aman Kumar Shrivastava

Eshan Arora

Shivam Rana

Aman Kumar Shrivastava

Manya

Barkha Arora

Manish Kumar

Shreya Shrivastava

Sharandeep Singh

Archit Segal

Aman Kumar Shrivastava

Hridey Preet

Ayan Ghosh