If sin s =2/3
and cos s =
√5
over
3

what are the values of the remaining four trigonometric
functions of s?

Question

If sin s =2/3
and cos s =
√5
over 
3

what are the values of the remaining four trigonometric
functions of s?

Abigail Armstrong · Accepted Answer

As below

Given #sin s = a / c = 2/3#

#:. a = 2, c = 3#

Similarly, #cos s = b / a = sqrt5 / 3#

#:. b = sqrt5, c = 3#

Verification :

As per Pythagoras theorem,

#c^2 = a^2 + b^2#

L H S = #c^2 = 3^2 = 9#

R H S = #a^2 + b^2 = 2^2 + (sqrt5)^2 = 4 + 5 = 9#

Hence proved.

Other trigonometric functions :

#tan s = a / b = 2 / sqrt5#

#csc s = 1 / sin s = 1 / (2/3) = 3/2#

#sec s = 1 / cos s = 1 /(sqrt5/3) = 3 / sqrt5#

#cot s = 1 / tan s = 1 / (2 / sqrt5) = sqrt5 / 2#

Aaliyah Abdullah · Answer

undefined Given that sin(s) = 2/3 and cos(s) = √5/3, we can use the Pythagorean identity to find the value of cos(s) and then determine the remaining trigonometric functions.

1. **sin(s) = 2/3**: Given.
2. **cos(s) = √5/3**: Given.
3. **tan(s) = sin(s)/cos(s)**: Substitute the given values into the formula to find tan(s).
   tan(s) = (2/3) / (√5/3) = 2/√5.
4. **csc(s) = 1/sin(s)**: Reciprocal of sin(s).
   csc(s) = 1 / (2/3) = 3/2.
5. **sec(s) = 1/cos(s)**: Reciprocal of cos(s).
   sec(s) = 1 / (√5/3) = 3/√5.
6. **cot(s) = 1/tan(s)**: Reciprocal of tan(s).
   cot(s) = 1 / (2/√5) = √5/2.

If sin s =2/3 and cos s = √5 over 3 what are the values of the remaining four trigonometric functions of s?