Use the trigonometric identities to find sin x tan x and cot

Use the trigonometric identities to find sin x, tan x, and cot(-x), given cos x = 5/4 and x in quadrant IV. Given tan x = - 5/4. where pi/2

Solution:

given sinx=-1/2

x and y an third quadrant .tan and cot are positive.rest all trigonometric ratios are negative

sinx=opp side /hypotenuse.=-1/2

hyp² =opp² + adj ²

adj ² =hyp² -opp²

⁼2² -1²

=4-1=3

adj= 3

cosx= -3/2

tanx=sinx/cosx=-1/2/-3/2=1/3

cotx=1/tanx=3

cosy=-2/5

cosy=adj/hyp=-2/5

adj=2 hyp=5

hyp² =adj² +opp²

opp² = hyp² -adj² =5² -2² 25-4=21

opp=21

siny=opp/hyp=-21/5

tany=siny/cosy=(21/5)/2/5

=21/2

coty=1/tany=2/21

sin(x+y)=sinxcosy+cosxsiny

=(-1/2)(-2/5)+ (-3/2)-21/5

=1/5+63/10

=2+63/10

cos(x-y)=cosxcosy+sinxsiny

= -(3/2).(-2/5)+(-1/2)(-21/5)

=3/5+21/10

=23+21/10

take 3 common

3(2+7)/10

tan(x+y)=tanx+tany/(1-tanxtany)

=(1/3)+(21/2)/1-(1/3)(21/2)