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13. If

    \[\mathbf{sec}\text{ }\mathbf{\theta }\text{ }=\text{ }\mathbf{13}/\mathbf{5},\]

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CBSE | Class 10 | Maths | RD Sharma | Trigonometric Ratios
13. If

    \[\mathbf{sec}\text{ }\mathbf{\theta }\text{ }=\text{ }\mathbf{13}/\mathbf{5},\]

show that

Solution:

Given,

    \[sec\text{ }\theta \text{ }=\text{ }13/5\]

We know that,

    \[sec\text{ }\theta \text{ }=\text{ }1/\text{ }cos\text{ }\theta \]

    \[\Rightarrowcos\text{ }\theta \text{ }=\text{ }1/\text{ }sec\text{ }\theta \text{ }=\text{ }1/\text{ }\left( 13/5 \right)\]

    \[\therefore cos\text{ }\theta \text{ }=\text{ }5/13\text{ }\ldots \ldots .\text{ }\left( 1 \right)\]

By definition,

    \[cos\text{ }\theta \text{ }=\]

adjacent side/ hypotenuse

    \[\ldots ..\text{ }\left( 2 \right)\]

Comparing

    \[\left( 1 \right)\text{ }and\text{ }\left( 2 \right),\]

we have

Adjacent side

    \[=\text{ }5\]

and hypotenuse

    \[=\text{ }13\]

By Pythagoras theorem,

Opposite side = √((hypotenuse) 2 – (adjacent side)2)

    \[=\text{ }\surd ({{13}^{2}}~\text{ }{{5}^{2}})\]

    \[=\text{ }\surd \left( 169\text{ }\text{ }25 \right)\]

    \[=\text{ }\surd \left( 144 \right)\]

    \[=\text{ }12~~~~~~\]

Thus, opposite side

    \[=\text{ }12\]

By definition,

    \[tan\text{ }\theta \text{ }=\]

opposite side/ adjacent side

    \[\therefore tan\text{ }\theta \text{ }=\text{ }12/\text{ }5\]

From, let’s divide the numerator and denominator by

    \[cos\text{ }\theta .\]

We get,

    \[\Rightarrow \left( 2\left( 12/5 \right)\text{ }\text{ }3 \right)\text{ }/\text{ }\left( 4\left( 12/5 \right)\text{ }\text{ }9 \right)\]

[using the value oftan θ]

    \[\Rightarrow \left( 24\text{ }\text{ }15 \right)\text{ }/\text{ }\left( 48\text{ }\text{ }45 \right)\]

[After taking Ltan θCM and cancelling it]

    \[\Rightarrow 9/3\text{ }=\text{ }3\]

    \[\therefore =\text{ }3\]

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