Trigonometric Functions

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Math › Trigonometric Functions

Questions 1 - 4

1

Simplify the function below:

$f(x)=cos^2(\frac{1}{2}cos^{-1}x)$

$\frac{1+x}{2}$

$\frac{1-x}{2}$

$\frac{x}{2}$

Explanation

We need to use the following formulas:

a) $cos^2x=\frac{1+cos2x}{2}$

and

b) $cos(cos^{-1}x)=x$

We can simplify as follows:

$f(x)=cos^2(\frac{1}{2}cos^{-1}x)=\frac{1+cos(cos^{-1}x)}{2}=\frac{1+x}{2}$

2

Simplify the function below:

$f(x)=cos^2(\frac{1}{2}cos^{-1}x)$

$\frac{1+x}{2}$

$\frac{1-x}{2}$

$\frac{x}{2}$

Explanation

We need to use the following formulas:

a) $cos^2x=\frac{1+cos2x}{2}$

and

b) $cos(cos^{-1}x)=x$

We can simplify as follows:

$f(x)=cos^2(\frac{1}{2}cos^{-1}x)=\frac{1+cos(cos^{-1}x)}{2}=\frac{1+x}{2}$

3

Which of the following is not in the range of the function $y = \sin{x}$ ?

$\frac{3}{2}$

$-\frac{1}{2}$

$\frac{3}{4}$

$\frac{2}{3}$

Explanation

The range of the function $y= \sin x$ is all numbers between and (the sine wave never goes above or below this).

Of the choices given, $\frac{3}{2}$ is greater than and thus not in this range.

4

Which of the following is not in the range of the function $y = \sin{x}$ ?

$\frac{3}{2}$

$-\frac{1}{2}$

$\frac{3}{4}$

$\frac{2}{3}$

Explanation

The range of the function $y= \sin x$ is all numbers between and (the sine wave never goes above or below this).

Of the choices given, $\frac{3}{2}$ is greater than and thus not in this range.

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