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 $large frac{13; cos;theta-12;cot;theta}{13;cos;theta+12;cot;theta}$
A. $dpi{110}frac{9}{13}$	B. $dpi{110}frac{5}{13}$	C. 0	D. None of these
Right Option : C
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Question : 2

If $3;sec; theta = 5$ find the value of $frac{5;sin;theta-4;cot;theta}{5;sin;theta+4;cot;theta}$
A. $dpi{110}frac{9}{3}$	B. $dpi{110}frac{1}{7}$	C. 0	D. None of these
Right Option : B
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Question : 3

Find the value of $sin;C$ when it is given that $cot; A = frac{a}{b}$
A. $dpi{110}frac{b}{sqrt{a^2-b^2}}$	B. $dpi{110}frac{a}{sqrt{a^2-b^2}}$	C. $dpi{110}frac{a}{sqrt{a^2+b^2}}$	D. None of these
Right Option : C
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