Question

If sine of x equals 1 over 2, what is cos(x) and tan(x)? Explain your steps in complete sentences.

Answer 1

The value of sin(x) is given to be 1/2 which is equal to 0.5.

Part 1) Finding Cos(x)

We can find the value of cos(x) using the Pythagorean Identity, which states:


`sin^(2)(x) +cos^(2)(x) =1 \\ \\ cos^(2)(x)=1-sin^(2)(x) \\ \\ cos(x)=+- \sqrt{1-sin^(2)(x)}`

Using the value of sin(x) in above equation, we get:


`cos(x)=+- \sqrt{1-(0.5)^(2) }=+- ( √(3) )/(2)`

The positive value of cos(x) indicates that the terminal arm of angle x lies in 1st quadrant, and the negative value of cos(x) indicates that the terminal arm lies in 2nd quadrant as in 2nd quadrant sin is positive and cos is negative.

Part 2) Finding Value of Tan(x)

tan(x) is the ratio of sin(x) and cos(x).

So, if x lies in 1st quadrant, tan(x) will be:


`tan(x)= (0.5)/( ( √(3) )/(2) ) \\ \\ = (1)/( √(3) ) \\ \\ = ( √(3) )/(3)`

And if x lies in 2nd quadrant, the value of tan(x) will be:


`tan(x)= (0.5)/( -( √(3) )/(2) ) \\ \\ = -(1)/( √(3) ) \\ \\ = - ( √(3) )/(3)`