Question

It is given that tanθ=0.75 and that π3π2. David says he can use the Pythagorean identity to find the exact value instead of an approximation of sinθ and cosθ. Is David correct? Explain.

Answer 1

David is correct. The Pythagorean identity can be used to find the exact values of sinθ and cosθ when tanθ is given.

Step-by-step explanation:

David is correct. The Pythagorean identity states that sin^2θ + cos^2θ = 1. Using the given information that tanθ = 0.75, we can find sinθ and cosθ. Since tanθ = sinθ/cosθ, we have sinθ/cosθ = 0.75. Rearranging the equation gives sinθ = 0.75cosθ. Squaring both sides gives sin^2θ = 0.5625cos^2θ. Substituting this into the Pythagorean identity gives 0.5625cos^2θ + cos^2θ = 1. Solving for cos^2θ, we get cos^2θ = 1/(0.5625+1) = 0.36. Taking the square root of both sides gives cosθ = 0.6. Substituting the value of cosθ into the equation sinθ = 0.75cosθ gives sinθ = 0.75(0.6) = 0.45.