Which of the following would be an acceptable first step in simplifying the expression $\frac{\sin x}{1+\sin x}$ ?
A. $\frac{1}{\csc x+\sin x}$
B. $\sin x+1$
C. $\frac{\sin x(1-\sin x)}{(1+\sin x)(1-\sin x)}$
Answer (1)
Analyze the given expression 1 + s i n x s i n x .
Check option A by dividing the numerator and denominator by sin x , resulting in c s c x + 1 1 , which does not match the option.
Check option B, which is sin x + 1 , and is not a valid manipulation.
Check option C by multiplying the numerator and denominator by ( 1 − sin x ) , resulting in ( 1 + s i n x ) ( 1 − s i n x ) s i n x ( 1 − s i n x ) , which matches the option.
The correct first step is option C: ( 1 + sin x ) ( 1 − sin x ) sin x ( 1 − sin x ) .
Explanation
Understanding the Problem We are given the expression 1 + s i n x s i n x and asked to identify an acceptable first step in simplifying it from the given options.
Analyzing the Options Let's analyze each option:
Option A: c s c x + s i n x 1 . To check if this is a valid first step, we can divide both the numerator and the denominator of the original expression by sin x . This gives us s i n x 1 + s i n x s i n x s i n x = s i n x 1 + 1 1 = c s c x + 1 1 . This does not match option A, so option A is incorrect.
Option B: sin x + 1 . This is not a valid algebraic manipulation of the original expression, so option B is incorrect.
Option C: ( 1 + s i n x ) ( 1 − s i n x ) s i n x ( 1 − s i n x ) . To check if this is a valid first step, we can multiply both the numerator and the denominator of the original expression by ( 1 − sin x ) . This gives us ( 1 + s i n x ) ( 1 − s i n x ) s i n x ( 1 − s i n x ) . This matches option C, so option C is correct.
Conclusion Therefore, the correct first step in simplifying the expression is to multiply both the numerator and the denominator by ( 1 − sin x ) .
Examples
Simplifying trigonometric expressions is useful in various fields such as physics, engineering, and computer graphics. For example, in physics, simplifying expressions involving trigonometric functions can help in analyzing wave phenomena, such as light or sound waves. In computer graphics, simplifying trigonometric expressions can help in optimizing calculations for rendering images or animations.
