Question

If 5 sin theta =3,then find cos theta , tan theta, sec theta-cosec theta

Answer 1

$\sf\large\underline{Given:}$

$\sf{\implies 5sin\theta=3}$

$\sf\large\underline{To\: Find:}$

$\sf{\implies cos\theta=?}$

$\sf{\implies tan\theta=?}$

$\sf{\implies sec\theta-cosec\theta=?}$

$\sf\large\underline{Solution:}$

• In right angle triangle ABC in which AB=3 AC=5 and angle B=90° BC=?. Here we use Pythagoras theorem to calculate the length of BC. Just notice once on given attachment.

✮ Pythagoras formula used✮

$\sf\small{\implies base=\sqrt{hypotenous^2-perpendicular^2}}$

$\tt{\implies 5sin\theta=3\implies sin\theta=\dfrac{3}{5}}$

$\rm{\implies BC=\sqrt{AC^2-AB^2}}$

$\rm{\implies BC=\sqrt{5^2-3^2}}$

$\rm{\implies BC=\sqrt{25-9}}$

$\rm{\implies BC=\sqrt{16}=4}$

✮ By calculating we get here ✮

$\bf{\implies AB=3\:\:,BC=4\:\:,AC=5}$

• ✮ Now calculate value here ✮

$\rm{\implies cos\theta=\dfrac{BC}{AC}=\dfrac{4}{5}}$

$\rm{\implies tan\theta=\dfrac{AB}{BC}=\dfrac{3}{4}}$

$\bf{\implies sec\theta-cosec\theta}$

$\rm{\implies \dfrac{AC}{BC}-\dfrac{AC}{AB}}$

$\rm{\implies \dfrac{5}{4}-\dfrac{5}{3}}$

$\rm{\implies \dfrac{15-20}{12}}$

$\rm{\implies \dfrac{-5}{12}}$

$\sf\large{Hence,}$

$\sf{\implies cos\theta=\dfrac{4}{5}}$

$\sf{\implies tan\theta=\dfrac{3}{4}}$

$\sf{\implies sec\theta-cosec\theta=\dfrac{-5}{12}}$

Answer 2

$\bigstar$ $\sf{\purple{\underline{\underline{\blue{To\:Find:-}}}}}$

• $\tt{\cos{\theta}\:=\:?}$

• $\tt{\tan{\theta}\:=\:?}$

• $\tt{\sec{\theta}\:-\:\csc{\theta}\:=\:?}$

$\bigstar$ $\sf{\blue{\underline{\underline{\purple{SOLUTION:-}}}}}$

☞ GIVEN :-

• $\tt{5\:\sin{\theta}\:=\:3\:}$

☞ See the attachment picture :-

• There,

1. sinθ = p/h
2. cosθ = b/h
3. tanθ = p/b
4. cotθ = b/p
5. secθ = h/b
6. cosecθ = h/p

☞ FORMULA :-

$\tt{\underline{\green{\boxed{h^2\:=\:p^2\:+\:b^2}}}}$

☞ CALCULATION :-

$\tt{\:\:\:\:\:{\implies\:\sin{\theta}\:=\:{\dfrac{3}{5}}\:}}$

☞ From the above equation,

• sinθ = 3/5 = p/h

1. p = 3
2. h = 5

$\tt{\underline{\blue{\boxed{b^2\:=\:h^2\:-\:p^2}}}}$

$\tt{\implies\:b\:=\:{\sqrt{h^2\:-\:p^2}}\:}$

$\tt{\implies\:b\:=\:{\sqrt{5^2\:-\:3^2}}\:}$

$\tt{\implies\:b\:=\:{\sqrt{(25\:-\:9)}}\:}$

$\tt{\implies\:b\:=\:{\sqrt{16}}\:}$

$\tt{\implies\:b\:=\:4}$

• So, b = 4

☞ Now, Calculate all the values .

1) $\checkmark\:\tt{\boxed{\cos{\theta}\:=\:{\dfrac{b}{h}}\:}}$

$\tt{\implies\:{\cos{\theta}\:=\:{\dfrac{4}{5}}\:}}$

2) $\checkmark\:\tt{\boxed{\tan{\theta}\:=\:{\dfrac{p}{b}}\:}}$

$\tt{\implies\:{\tan{\theta}\:=\:{\dfrac{3}{4}}\:}}$

3) $\checkmark\:\tt{\boxed{\sec{\theta}\:=\:{\dfrac{h}{b}}\:}}$

$\tt{\implies\:{\sec{\theta}\:=\:{\dfrac{5}{4}}\:}}$

4) $\checkmark\:\tt{\boxed{\csc{\theta}\:=\:{\dfrac{h}{p}}\:}}$

$\tt{\implies\:{\csc{\theta}\:=\:{\dfrac{5}{3}}\:}}$

5) $\checkmark\:\tt{\boxed{\sec{\theta}\:-\:\csc{\theta}\:}}$

$\tt{\:=\:{\dfrac{5}{4}}\:-\:{\dfrac{5}{3}}\:}$

$\tt{\:=\:-\:{\dfrac{5}{12}}\:}$