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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.

123 Views

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 $fraction numerator sinθ plus cosθ over denominator sinθ minus cosθ end fraction ?$

Let us draw a right angle triangle, right angled at B.
We know that

tan squared straight theta space equals space 4 over 3 equals AB over AC

Let,AB -4K. BC -3K
Where K is positive number.
Using pythagoras theorem, we have
⇒ AC² = AB² + BC²= (4K)² + (3K)²= 16K² + 9K²= 25 K²So AC = 5K
Now, AB = 4K, BC = 3K and AC = 5K
Therefore,
Let us draw a right angle triangle, right angled at B.We know thatLet

Let us draw a right angle triangle, right angled at B.We know thatLet

fraction numerator sinθ plus cosθ over denominator sinθ minus cosθ end fraction space equals space fraction numerator begin display style AB over AC end style plus begin display style BC over AC end style over denominator begin display style AB over AC end style minus begin display style BC over AC end style end fraction

equals space fraction numerator begin display style fraction numerator 4 straight K over denominator 5 straight K end fraction end style plus begin display style fraction numerator 3 straight K over denominator 5 straight K end fraction end style over denominator begin display style fraction numerator 4 straight K over denominator 5 straight K end fraction end style minus begin display style fraction numerator 3 straight K over denominator 5 straight K end fraction end style end fraction space equals space fraction numerator begin display style 4 over 5 end style plus begin display style 3 over 5 end style over denominator begin display style 4 over 5 end style minus begin display style 3 over 5 end style end fraction space equals space fraction numerator begin display style fraction numerator 4 plus 3 over denominator 5 end fraction end style over denominator begin display style fraction numerator 4 minus 3 over denominator 5 end fraction end style end fraction space equals space fraction numerator 7 minus 4 over denominator 1 end fraction

142 Views

Short Answer Type

86.

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

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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

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