If cot θ = evaluate: from Mathematics Introduction to Tr

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Introduction to Trigonometry

Long Answer Type

1. In ΔABC, right angled at B. AB = 24 cm, BC = 7 cm. Determine:
(i) sin A cos A,
(ii) sin C, cos C.

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2. In the given figure, find tan P - cot R.

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Short Answer Type

3.

If $sinA space equals space 3 over 4 comma space$ calculate cos A and tan A.

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4. Given 15 cot A = 8, find sin A and sec A.

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Long Answer Type

5.

Given $space secθ space equals space 13 over 12 comma space$ calculate all other trigonometric ratios.

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6. If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B.

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7.
If cot θ = $7 over 8 comma$ evaluate:
$fraction numerator left parenthesis 1 plus sinθ right parenthesis space left parenthesis 1 minus sinθ right parenthesis over denominator left parenthesis 1 plus cosθ right parenthesis space left parenthesis 1 minus cosθ right parenthesis end fraction$

Consider a right angle triangle, right angled at B and ∠ACB = θ.
Here,

space cotθ space equals space BC over AB space equals space 7 over 8

BC = 7K
Let BC = 7K
AB = 8K
Using Pythagoras theorem, we have

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equals 64 straight K squared plus 49 straight K squared space equals space 113 straight K squared

rightwards double arrow

AC space equals space square root of 113 straight K

Consider a right angle triangle, right angled at B and ∠ACB = θ.He

Now,

sinθ space equals space AB over AC space equals space fraction numerator 8 straight K over denominator square root of 113 straight K end fraction equals fraction numerator 8 over denominator square root of 113 end fraction cosθ space equals space BC over AC space equals space fraction numerator 7 straight K over denominator square root of 113 straight K end fraction equals fraction numerator 7 over denominator square root of 113 end fraction

Now,

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space space equals space fraction numerator 1 minus open parentheses begin display style fraction numerator 8 over denominator square root of 113 end fraction end style close parentheses squared over denominator 1 minus open parentheses begin display style fraction numerator 7 over denominator square root of 113 end fraction end style close parentheses squared end fraction equals fraction numerator 1 minus begin display style 64 over 113 end style over denominator 1 minus begin display style 49 over 113 end style end fraction

equals space fraction numerator begin display style fraction numerator 113 minus 64 over denominator 113 end fraction end style over denominator begin display style fraction numerator 113 minus 49 over denominator 113 end fraction end style end fraction equals 49 over 64.

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Short Answer Type

8. If 3 cot A = 4, check whether

fraction numerator 1 minus tan squared straight A over denominator 1 plus tan squared straight A end fraction equals cos squared straight A space minus space sin squared straight A

or not.

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Long Answer Type

9. In triangle ABC, right angled at B, if

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find the value of :
(i) sin A cos C + cos A sin C
(ii) cos A .cos C - sin A.sin C.

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10. In ΔPQR, right angled at Q. PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.

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