Question

if sec theta=7 then evaluate cosec theta+tan theta+sin theta cosec theta+sin theta sec theta tan theta.

Answer 1

★${\red{\huge{\underline{\underline{\mathtt{Question:-}}}}}}$

If sec∅=7 ,then evaluate cosec ∅+tan∅+sin∅.cosec∅+sin∅ sec∅ tan∅.

★${\red{\huge{\underline{\underline{\mathtt{Solution:-}}}}}}$

$sec\theta = 7$

★ We know,

${\boxed{\green{\bold{sec\theta=\frac{Hypotenuse}{Base}=\frac{7}{1}}}}}$

Now, find the height.

Height ² + Base² = Hypotenuse ²

→ Height ²+ 1² = 7²

→ Height= $\sqrt48$

→ Height = $4\sqrt3$

Now,

${\boxed{\green{\bold{sin\theta=\frac{Height}{Hypotenuse}=\frac{4\sqrt3}{7}}}}}$

${\boxed{\green{\bold{cos\theta=\frac{Base}{Hypotenuse}=\frac{1}{7}}}}}$

${\boxed{\green{\bold{tan\theta=\frac{Height}{Base}=\frac{4\sqrt3}{1}}}}}$

${\boxed{\green{\bold{cosec\theta=\frac{Hypotenuse}{Height}=\frac{7}{4\sqrt3}}}}}$

$cosec\theta+tan\theta+sin\theta.cosec\theta+sin\theta.sc\theta.tan\theta$

= $\frac{7}{4\sqrt3}$+$\frac{4\sqrt3}{1}$+$\frac{4\sqrt3}{7}×\frac{7}{4\sqrt3}$+$\frac{4\sqrt3}{7}$×$7$×$\frac{4\sqrt3}{1}$

=$\frac{55}{4\sqrt3}+1+48$

=${\blue{\bold{\frac{55+196\sqrt3}{4\sqrt3}}}}$

★${\red{\huge{\underline{\underline{\mathtt{Answer:-}}}}}}$

${\blue{\bold{\frac{55+196\sqrt3}{4\sqrt3}}}}$

Answer 2

$\huge \mathfrak{Answer:-}$

$\implies$ (55 + 196√3)/4√3

$\large\underline\mathrm{Given:-}$

• sec(theta) = 7

$\large\underline\mathrm{To \: find}$

• cosec (theta) + tan (theta) + sin (theta) . cos (theta) + sin (theta) . sec (theta) . tan (theta).

$\large\underline\mathrm{Solution}$

$\implies$ sec (theta) = 7

$\implies$ sec (theta) = h/b = 7/1

Now, find the perpendicular.

$\implies$ p² + b² = h²

$\implies$ p² + 1² = 7²

$\implies$ p² = 7² - 1²

$\implies$ p² = 49 - 1

$\implies$ p² = 48

$\implies$ p = √48

$\implies$ p = 4√3

Now,

$\implies$ sin(theta) = p/h = (4√3)/7

$\implies$ cos(theta) = b/h = 1/7

$\implies$ tan(theta) = p/b = (4√3)/1

$\implies$ cosec(theta) = h/b = 7/(4√3)

$\implies$ cosec (theta) + tan (theta) + sin (theta) . cos (theta) + sin (theta) . sec (theta) . tan (theta).

$\implies$ 7/(4√3) + (4√3)/1 + (4√3)/7 × (4√3)/7 + (4√3)/7 × 7 × (4√3)/1

$\implies$ 55/(4√3) + 1 + 48

$\implies$ (55 + 196√3)/4√3

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