Use the given conditions to find the exact values of sin(2u), cos(2u), and tan(2u) using the double-angle formulas. tan(u) = 3/5, 0 < u < 𝜋/2

Answer 1

In Radian Measure, the arctangent of 3/5 (or 0.6) is u = 0.5404195003.

A calculator gives:

sin u as 0.5144957555; &

cos u as 0.8574929257.

Confirm that sin u/cos u (equal to tan u) does give 0.5144957555/0.8574929257 equal to 

0.6 as above.

Also note that 0.5404195003 falls between 0 and π/2 (equal to 1.570796327).

From sin (2u) = 2(sin u)(cos u), obtain 

sin (2 × 0.5404195003) = 2(sin 0.5404195003)(cos 0.5404195003).

Rewrite this last as sin (1.080839001) = 2 × 0.5144957555 × 0.8574929257

equal to 0.8823529412, which is supported by calculator.

From cos (2u) = cos²u − sin²u, obtain 

cos (1.080839001) = (0.8574929257)² − (0.5144957555)² equal to 0.4705882352,

which is supported by calculator.

Finally, tan (2u) is given by 2tan u / (1 − tan²u) 

or 2(0.6) / [1 − 0.6²] which simplifies to 1.875.

Note that 1.875 is also given by tan (2 × 0.5404195003) and by

sin (2 × 0.5404195003) over cos (2 × 0.5404195003).

Answer 2

tanu= 3/5

sinu = 3/sqr(9+25) = 3/sqr34

cosu =5/sqr34

sin2u = 2sinucosu = 2(3/sqr34)(5/sqr34)==30/34 =15/17

cos2u = cos^2(u) - sin^2(u) =25/34 -9/34= 16/34= 8/17

tan2u= sin2u/cos2u = 15/8 =1.875

or

tan2u = 2tanu/(1-tan^2(u) = 2(.6)/(1-.36) = 1.2/.64= 120/64= 30/16=15/8 = 1.875

those are all exact solutions, not rounded approximations

the following uses half angle formulas

tanu=3/5= .6= sinu/cosu

sinu= 3/sqr34=about .51

cosu=5/sqr34= about .86

sin(u/2)= sqr((1-cosu)/2) = .about .26

cos(u/2)= sqr((1+cosu)/2)= about .98

tan(u/2) = sin(u/2)/cos(u/2) =about .28

tan(u/2) = sinu/(1+cosu) = (1-cosu)/sinu = sqr((1-cosu)/(1+cosu)

pick one of the formulas, and it = about .98/.26 = about .28

all values are positive, as they're in quadrant I, so ignore negative signs in the half angle formulas

tan(u/2) = sinu/(1+cosu) = 3/sqr34/(5/sqr34+1)=3/sqr34/(5+sqr34)/sqr34 = 3/(5+sqr34)

= about .28

cos(u/2) = sqr((1+5/sqr34)/2) = about .98

sin(u/2) = sqr((1-5/sqr34)/2) = about .26

for exact values keep the square root signs,

but as a check on those exact answers find the approximate answers and see if they make sense