Question

If tan θ = 3/4 and 0^∘ < θ < 90^∘, evaluate:

a) sin θ

Answer 1

When tan θ = 3/4 and 0° < θ < 90°, by constructing a right-angled triangle with sides of 3 and 4, and calculating the hypotenuse using the Pythagorean theorem, we determine that sin θ = 3/5.

Step-by-step explanation:

If tan θ = 3/4 and 0° < θ < 90°, we need to find sin θ. To do this, we can think of a right-angled triangle where the opposite side to angle θ has length 3, and the adjacent side has length 4. According to the Pythagorean theorem, the hypotenuse of this triangle will be the square root of the sum of the squares of the other two sides: √(3² + 4²) = √(9 + 16) = √25 = 5.

Now, we use the definition of sin θ as the ratio of the opposite side to the hypotenuse. Thus,

sin θ = opposite/hypotenuse = 3/5.

The value of sin θ when tan θ = 3/4 in the given range of θ is 3/5.