Question

Please help!!

If θ is an angle in standard position whose terminal side passes through (3, 4), evaluate tan1/2θ
1/4
3/10
1/2
4/5

Answer 1

1/2 is the answer (actually, it's ±1/2)

Explanation:

The identity you need here is for a half angle of tangent. That identity is as follows:

`tan((\theta )/(2))=`±
`\sqrt{(1-cos\theta )/(1+cos\theta ) }`

If we need the cos of that angle, we need to find the missing hypotenuse. Applying Pythagorean's Theorem to that right triangle, we get that the hypotenuse is 5. The cos of the angle is 3/5. Filling in the formula, using only the principle root since you have not allowed for the negative in the choices you gave:

`tan((\theta )/(2))=\sqrt{(1-(3)/(5) )/(1+(3)/(5) ) }`

Turning each one of those 1's into 5/5 we combine the fractions to simplify to:

`tan((\theta )/(2))=\sqrt{((2)/(5) )/((8)/(5) ) }`

Bringing up the lower fraction and flipping to multiply gives us:

`tan((\theta )/(2))=\sqrt{(2)/(5)*(5)/(8)  }`

Canceling out the 5's and reducing the remaining fraction gives us

`tan((\theta )/(2))=\sqrt{(1)/(4) }`

and since the square root of 4 is 2, we end with a solution of 1/2