Trigonometric ratios for 45 degrees


 
 
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Trigonometric ratios for 45 degrees

Trigonometric Ratios for 45 Degrees: Value for the Trigonometrical Ratios for certain angle such as 30^0,60^090^0 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 Asin Acos Atan Acosec Asec Acot A
large 45^{circ}large frac{1}{sqrt{2}}large frac{1}{sqrt{2}}1large {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

Right Option : D
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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

Right Option : A
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Explanation
Question : 3

Solve the given expression :

sin;45^0+cos;45^0+(2+tan ;45^0+cot;45^0)

Right Option : D
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Explanation
 
 
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