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Trigonometric Functions

Short Answer Type

581.

Find the general expression for all angles whose sine is zero.

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582. Find the general expression for all angles whose tangent is zero.

119 Views

583.

Find the general solution or the general expression for all angles, whose cosine is zero.

110 Views

584.

Find the general expression for all the angles whose cotangent is zero.

102 Views

585. Find a general expression for all angles having the same sine, i.e. when sin x = sin y, find the general value of x

We have sinx = siny
$rightwards double arrow$ sinx - siny = 0

$rightwards double arrow$ $space space 2 cos open parentheses fraction numerator straight x plus straight y over denominator 2 end fraction close parentheses space sin open parentheses fraction numerator straight x minus straight y over denominator 2 end fraction close parentheses space equals space 0$

$rightwards double arrow$ either $cos open parentheses fraction numerator straight x plus straight y over denominator 2 end fraction close parentheses equals 0 space or space sin fraction numerator straight x minus straight y over denominator 2 end fraction equals 0$

$rightwards double arrow$ either $fraction numerator straight x plus straight y over denominator 2 end fraction equals left parenthesis 2 straight k plus 1 right parenthesis straight pi over 2 comma space straight k element of straight Z space space space space space space space$ or $fraction numerator straight x minus straight y over denominator 2 end fraction equals kπ comma space straight k space element of space straight Z$

$rightwards double arrow$ either $straight x plus straight y equals left parenthesis 2 straight k plus 1 right parenthesis straight pi comma space straight k space element of space straight Z$ or $space space straight x minus straight y space equals space 2 kπ comma space straight k space element of space straight Z$

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$rightwards double arrow$ $straight x equals nπ plus left parenthesis negative 1 right parenthesis to the power of straight n straight y comma space straight n space element of space straight Z$

The general solution is: $nπ plus left parenthesis negative 1 right parenthesis to the power of straight n straight y comma space straight n space element of space straight Z$

COROLLARY: If cosecx = cosec y, then six = siny

So, $straight x equals nπ plus left parenthesis negative 1 right parenthesis to the power of straight n straight y comma space where space straight n space element of space straight Z$