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Introduction to Trigonometry

Short Answer Type

81. If sin θ = cos θ, find the value of θ.

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82. If tan θ = cot (30° + θ), find the value of θ.

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83. If sin 3θ = cos (θ - 6°) and θ-6° are acute angles, find the value of θ.

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Long Answer Type

84. Given that

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what is the value of

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85.

If $tanθ space equals space 4 over 3 comma$ what is the value of $fraction numerator sinθ plus cosθ over denominator sinθ minus cosθ end fraction ?$

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Short Answer Type

86.
If $space tanA space equals space 5 over 12 comma$ find the value of (sinA + cosA) secA.

Let us consider a right triangle right angled at B.
We have,

rightwards double arrow space space tanA space equals space 5 over 12 space equals space AB over BC

If AB = 5K, BC = 12K
Then,

space AC squared equals AB squared plus BC squared space equals space left parenthesis 5 straight K right parenthesis squared plus left parenthesis 12 straight K right parenthesis squared space space space space space space space space space space equals space 25 straight K squared plus 144 straight K squared space equals space 169 straight K squared

So, AC = 13K
Now,

sinA space equals space AB over AC equals fraction numerator 5 straight K over denominator 13 straight K end fraction equals 5 over 13 cosA space equals space BC over AC equals fraction numerator 12 straight K over denominator 13 straight K end fraction space equals 12 over 13

secA space equals space AC over BC equals fraction numerator 13 straight K over denominator 12 straight K end fraction equals 13 over 12

Therefore,

left parenthesis sinA plus cosA right parenthesis space secA space equals space open parentheses 5 over 13 plus 12 over 13 close parentheses cross times 13 over 12 equals space open parentheses fraction numerator 5 plus 12 over denominator 13 end fraction close parentheses cross times 13 over 12 space equals space 17 over 13 cross times 13 over 12 space equals space 17 over 12.

89 Views

87.

If $cosA space equals space 7 over 25 comma$ find the value of (tanA + cotA).

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88.

If $cotθ space equals space 15 over 8 comma$ then evaluate "<pre

95 Views

89.

If $secA space equals space 15 over 7 space and space straight A plus straight B space equals space 90 degree comma space$ find the value of cosec B.

111 Views

90.

Evaluate:
$fraction numerator sec left parenthesis 90 degree minus straight theta right parenthesis times cosecθ space minus space tan left parenthesis 90 degree minus straight theta right parenthesis space cotθ plus left parenthesis cos squared 35 degree plus cos squared 55 degree right parenthesis over denominator tan 5 degree space tan space 15 degree space tan space 45 degree space tan space 75 degree space tan space 85 degree end fraction.$