Question

Simplify the expression 9 − x² as much as possible after substituting 3 sin(θ) for x. (Assume 0° < θ < 90°.)

a) 9 - 9 sin² (θ)
b) 9 - 9 cos²(θ)
c) 9 - 9 sin (θ)
d) 9 - 27 sin (θ)

Answer 1

After substituting 3 sin(θ) for x in the expression 9 - x² and using the Pythagorean trigonometric identity, the simplified form of the expression is 9 cos²(θ), corresponding to answer choice (b).

Step-by-step explanation:

To simplify the expression 9 − x² after substituting 3 sin(θ) for x, where 0° < θ < 90°, you perform the substitution directly into the expression. When you substitute, you square the term 3 sin(θ) to get 9 sin²(θ), and the expression becomes 9 − 9 sin²(θ). However, using the Pythagorean identity, we know that sin²(θ) + cos²(θ) = 1, which can be rearranged to cos²(θ) = 1 - sin²(θ). Therefore, by substituting, we see that 9 - 9 sin²(θ) is equivalent to 9 - 9(1 - cos²(θ)), which simplifies further to 9 - 9 + 9 cos²(θ) or simply 9 cos²(θ). The simplified form of the expression after the substitution is, therefore, 9 cos²(θ), which corresponds to answer choice (b).

To simplify the expression 9 - x² after substituting 3 sin(θ) for x, we need to substitute the value of x into the expression and simplify. Since x = 3 sin(θ), we have:

9 - (3sin(θ))²

Expanding the square:

9 - 9sin²(θ)

Therefore, the correct answer is a) 9 - 9sin²(θ).