With a graphing calculator, plot Y1=sin((x)/(2)), Y2=(1)/(2) sinx, and Y3=√((1+cosx)/(2)) in the

step 1 you have to again use a graphing calculator over here. So we have sine x by 2, 1 by 2 sine x and

With a graphing calculator, plot Y1=sin((x)/(2)), Y2=(1)/(2)...

With a graphing calculator, plot $Y_{1}=\sin \left(\frac{x}{2}\right), Y_{2}=\frac{1}{2} \sin x$, and $Y_{3}=\sqrt{\frac{1+\cos x}{2}}$ in the same viewing rectangle $[0,2 \pi]$ by [-1,1] . Which graphs are the same?

Key Concepts

Trigonometric Identities (Half?Angle and Double?Angle Formulas)

Trigonometric identities, such as the half?angle formulas and double?angle formulas, are essential tools for rewriting and comparing trigonometric expressions. For example, identities can show that one function written in one form, like a square root expression, may be equivalent to an absolute value of a cosine function, or that products of sine and cosine can be rewritten in terms of a single trigonometric function. This understanding is crucial when studying relationships between different functions.

Absolute Value Considerations in Trigonometric Expressions

Sometimes, as when taking the square root of a squared trigonometric function, the resulting function is expressed in terms of an absolute value. This concept is important because the presence of an absolute value can change the behavior of the function—making it always nonnegative—and can lead to differences in the graph compared to the original trigonometric expression. Recognizing when an absolute value is implicit or explicit helps in correctly comparing and graphing functions.

Trigonometric Function Transformations

This concept involves understanding how changes to the input or output of a trigonometric function affect its graph. For example, replacing x with x/2 stretches the period of the sine graph, while multiplying the function by a constant (like 1/2) changes its amplitude. Recognizing these changes is key when comparing the shapes and positions of different trigonometric graphs.

Graphing of Trigonometric Functions

Graphing trigonometric functions requires an understanding of their amplitude, period, phase shift, and vertical shift. When multiple functions are plotted in the same coordinate system, it is important to examine these features to determine whether any of the graphs coincide or are merely similar in certain regions. This forms the basis for comparing and identifying equivalent or different functions visually.

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