Reciprocal Trigonometric Ratios


 
 
Concept Explanation
 

Reciprocal Trigonometric Ratios

Reciprocal Trignometric Ratios: These are the trigonometric ratios which are the reciprocal of the three ratios sin A, cos A and tan A. The ratios are cosec A, sec A and cot A are respectively, the reciprocals of the ratios sin A,cos A and tan A. For a right triangle ABC right angled at B we observe that

cosecant ;of;angle A=cosec; A=frac{1}{sine;of;angle A}=frac{hypotenuse}{side;opposite;to;angle;A}=frac{AC}{BC}

secant ;of ;angle A=sec;A=frac{1}{cosine;of;angle A}=frac{hypotenuse}{side;adjacent;to;angle;A}=frac{AC}{AB}

cotangent; of ;angle A=cot;A=frac{1}{tan;A}=frac{side;adjacent;to;angle;A}{side;opposite;to;angle;A}=frac{AB}{BC}

cot;A =frac{AB}{BC}=frac{frac{AB}{AC}}{frac{BC}{AC}}=frac{cos;A}{sin;A}

Illustration:  ABC is a right angled triangle. If AB = 21 cm, BC =  20 cm and CA  = 29 cm and large angle;A= theta. ;Find; cot;theta..

Solution: As per definition we know that

large cot;theta= frac{Adjacent; Side}{Opposite ;Side}=frac{AB}{BC} = frac{21}{20}

Illustration: Simplify the expression

cot; Aleft ( frac{1}{cosec A} right )+ sec;A- left ( frac{1}{sec; A} right )

Solution: To simplify the expression we will convert the ratios in terms of sin and cos.

cot; Aleft ( frac{1}{cosec A} right )+ sec;A- left ( frac{1}{sec; A} right )

=frac{cos; A}{sin;A}left (sin ;A right )+ frac{1}{cos;A}- cos;A

=cos; A+ frac{1}{cos;A}- cos;A=frac{1}{cos;A}= sec;A

Illustration: If  cosTheta =frac {p}{q}, find the value of cosecTheta .

Solution:

We know  cosTheta =frac {base}{hypotenuse}

We draw a right triangle, with right angle at C and having base BC = pk and hypotenuse BA = qk.

By the pythagorean theorem

                                                 AB^2=BC^2+AC^2

Rightarrow                                           (qk)^2=(pk)^2+AC^2

Rightarrow                                           AC^2=(qk)^2-(pk)^2 =k^2(q^2-p^2)

Rightarrow                                          AC=k sqrt {q^2-p^2}: : : : : : [therefore AC>0]

Thus,                           cosecTheta =frac {hypotenuse}{perpendicular }=frac {qk}{ksqrt {q^2-p^2}}=frac {q}{sqrt {q^2-p^2}}

 

Sample Questions
(More Questions for each concept available in Login)
Question : 1

If 13;sin ; theta = 12 Find the value of

large frac{13; cos;theta-12;cot;theta}{13;cos;theta+12;cot;theta}

Right Option : C
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Explanation
Question : 2

Express the trigonometric ratio tan A in terms of cot A.

Right Option : C
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Explanation
Question : 3

ABC is a right angled triangle.If AB = 21cm, BC= 20cm and CA  = 29cm and angle;A= theta. ;Find; 21;sec;theta -29;cos;theta..

Right Option : C
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Explanation
 
 
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