If sin (90- theta) = [the square root of 5] / 3, determine the value of tan theta

Question

multiple choice Answers,

a. 2/3

b. Square root of 5 / 5

c. square root of 5 / 2

d. 2 times square root of 5 / 5

Answer 1

Apply the difference of two angles identity:

 

     sin(90°-θ) = sin(90^o)cos(θ) - sin(θ)cos(90^o) = cos(θ) = (√5)/3

 

Now apply the Pythagorean Identity to find sin(θ):

 

     sin²(θ) + cos²(θ) = 1

     sin²(θ) + 5/9 = 1

     sin²(θ) = 4/9

     sin(θ) = 2/3

 

     tan(θ) = sin(θ)/cos(θ) = (2/3) / ((√5)/3) = (2/3)·(3/(√5)) = 2/√5

 

Rationalizing the denominator:

 

     tan(θ) = (2/√5)·(√5/√5) = (2√5)/5