If sin𝜃 = 2/3 and cos𝜃 > 0, then find tan𝜃.

Answer 1

Since sin(θ) > 0 and cos(θ) > 0, θ must be in quadrant 1.

The length of the hypotenuse of the triangle formed is 3.

The length of the adjacent side of the triangle is √(3² - 2²) = √5.

The tan(θ) = side opposite / side adjacent = 2 / √5

Answer 2

Since sin θ = opposite / hypotenuse, we can think of a triangle with opposite side length 2 and hypotenuse length 3. Now we can use the Pythagorean Theorem to find the adjacent side length.

2^2 + X^2 = 3^2

4 + X^2 = 9

X^2 = 5

X = √5

[Note that we can ignore the possible solution of -√5 since a length cannot be negative]

Now we can find tan θ = opposite / adjacent

tan θ = 2 / √5

To rationalize, just multiply the numerator and denominator by √5.

tan θ = 2√5 / √5√5 = 2√5 / 5