What is sin(x), cos(x), tan(x), csc(x), sec(x), and cot(x) if sin(2x)=5/13 and x is in between 0 and 45 degrees?

Answer 1

I'm assuming you need exact answers.

If sin(2x) = 5/13 then cos(2x) = 12/13

[Note that we get cos(2x) either by drawing the 5-12-13 right triangle

or from the identity cos²(2x) = 1 – sin²(2x) = 1 – 25/169 = 144/169]

We know that our angle is in the first quadrant, so all six trig functions will have positive values.

We can use half-angle formulas to find sin(x) and cos(x)

sin(x) = √[(1 – cos(2x))/2] = √[(1 – 12/13)/2] = √[(1/13)/2] = 1/√26

cos(x) = √[(1 + cos(2x))/2] = √[(1 + 12/13)/2] = √[(25/13)/2] = 5/√26

Check: sin²(x) + cos²(x) = 1/26 + 25/26 = 1

tan(x) = sin(x)/cos(x) = (1/√26)/(5/√26) = 1/5

csc(x) = 1/sin(x) = √26

sec(x) = 1/cos(x) = (√26)/5

cot(x) = 1/tan(x) = 5

Answer 2

for exact answers

sin(x)=(1/2)[+/-√(1+sin(2x))+/-√(1-sin(2x))]

given sin(2x)=5/13 and 0<2x<pi/2

sin(x)=(1/2)[√(18/13)-√(8/13)]

yay

sin(2x)=2sin(x)cos(x)

cos(x)=sin(2x)/{2sinx)

typo and algebra fixed

cos(x)=(5/26)/(√(18/13)-√(8/13))

yay,

and

tan(x)=(26/20)(2/13)

=1/5

Answer 3

u = 2x

sin u = 5 / 13

sin^-1 5/13 = u

22.61986495 = u

22.61986495 = 2x

11.3 = x

Can you now calculate the values of the six trigonometric functions?