Question

if theta and phi lie in the first quadrant such that sin theta=8/17 and cos phi=12/13. Find the value of
1) sin(theta-phi)​
2) cos(theta+phi)
3)tan(theta-phi)

Answer 1

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

Given that,

$\rm :\longmapsto\:0 < \theta < \dfrac{\pi}{2}$

$\rm :\longmapsto\:0 < \phi < \dfrac{\pi}{2}$

$\red{\rm :\longmapsto\:sin\theta = \dfrac{8}{17}}$

$\red{\rm :\longmapsto\:cos\phi = \dfrac{12}{13}}$

Now,

As it is provided that

$\rm :\longmapsto\:0 < \theta, \: \phi < \dfrac{\pi}{2}$

So,

$\rm :\longmapsto\:sin\theta ,cos\theta ,tan\theta ,cos\phi ,sin\phi ,tan\phi > 0$

Now,

We know that

$\red{\rm :\longmapsto\: {sin}^{2}\theta + {cos}^{2}\theta = 1}$

$\rm :\longmapsto\:\dfrac{64}{289} + {cos}^{2} \theta = 1$

$\rm :\longmapsto\: {cos}^{2}\theta = 1 - \dfrac{64}{289}$

$\rm :\longmapsto\: {cos}^{2}\theta = \dfrac{289 - 64}{289}$

$\rm :\longmapsto\: {cos}^{2}\theta = \dfrac{225}{289}$

$\rm :\longmapsto\: {cos}^{2}\theta = \dfrac{ {15}^{2} }{ {17}^{2} }$

$\bf\implies \:cos\theta = \dfrac{15}{17}$

Also, we know that,

$\rm :\longmapsto\:tan\theta = \dfrac{sin\theta }{cos\theta } = \dfrac{8}{17} \div \dfrac{15}{17} = \dfrac{8}{15}$

Also,

$\rm :\longmapsto\: {sin}^{2}\phi + {cos}^{2}\phi = 1$

$\rm :\longmapsto\: {sin}^{2}\phi + \dfrac{144}{169} = 1$

$\rm :\longmapsto\: {sin}^{2}\phi = 1 - \dfrac{144}{169}$

$\rm :\longmapsto\: {sin}^{2}\phi = \dfrac{169 - 144}{169}$

$\rm :\longmapsto\: {sin}^{2}\phi = \dfrac{25}{169}$

$\rm :\longmapsto\: {sin}^{2}\phi = \dfrac{ {5}^{2} }{ {13}^{2} }$

$\bf\implies \:sin\phi = \dfrac{5}{13}$

Also,

$\rm :\longmapsto\:tan\phi = \dfrac{sin\phi }{cos\phi } = \dfrac{5}{13} \div \dfrac{12}{13} = \dfrac{5}{12}$

Consider 1.

$\red{\rm :\longmapsto\:sin(\theta - \phi )}$

$\rm \: = \: \: sin\theta cos\phi - sin\phi cos\theta$

$\rm \: = \: \: \dfrac{8}{17} \times \dfrac{12}{13} - \dfrac{5}{13} \times \dfrac{15}{17}$

$\rm \: = \: \: \dfrac{96}{221} - \dfrac{75}{221}$

$\rm \: = \: \: \dfrac{96 - 75}{221}$

$\rm \: = \: \: \dfrac{21}{221}$

Consider:- 2

$\red{\rm :\longmapsto\:cos(\theta + \phi )}$

$\rm \: = \: \: cos\theta cos\phi - sin\theta sin\phi$

$\rm \: = \: \: \dfrac{15}{17} \times \dfrac{12}{13} - \dfrac{8}{17} \times \dfrac{5}{13}$

$\rm \: = \: \: \dfrac{180}{221} - \dfrac{40}{221}$

$\rm \: = \: \: \dfrac{180 - 40}{221}$

$\rm \: = \: \: \dfrac{140}{221}$

Consider :- 3

$\red{\rm :\longmapsto\:tan(\theta - \phi )}$

$\rm \: = \: \: \dfrac{tan\theta - tan\phi }{1 + tan\theta \: tan\phi }$

$\rm \: = \: \: \dfrac{\dfrac{8}{15} - \dfrac{5}{12} }{1 + \dfrac{8}{15} \times \dfrac{5}{12} }$

$\rm \: = \: \: \dfrac{96 - 75}{180 + 40}$

$\rm \: = \: \: \dfrac{21}{220}$

Answer 2

Answer:

θ

and ϕ

lie in the first quadrant, such that sinθ=817

and cosϕ=1213

, find the values of cos(θ+ϕ)

.