Trigonometric Functions and Graphs

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Trigonometry › Trigonometric Functions and Graphs

Questions 1 - 10

Which of the following is positive?

$\cos 45$

$\cos 90$

$\cos 135$

$\cos 180$

Explanation

When drawn from the origin, a line 45 degrees above (counterclockwise from) the positive x-axis lies in quadrant I. Cosine is defined as the ratio between the adjacent side of a triangle and the hypotenuse of the triangle. A right triangle can be drawn in quadrant I composed of any point on that line, the origin and a point on the x-axis. The hypotenuse of this triangle is considered a length, and is therefore positive. The adjacent side of this triangle lies along the positive x-axis. Since the adjacent side and hypotenuse are both represented by positive numbers, the fraction A/H is positive. Therefore, cos 45 is positive.

Solve the following equation by squaring both sides: $1 + cot(\theta) = csc(\theta)$

$\theta = 0, \pi$

$\theta = 0$

$\theta = \frac{\pi}{2}$

$\theta = \frac{\pi}{2}, \frac{3\pi}{2}$

Explanation

We begin with our original equation:

$1 + cot(\theta) = csc(\theta)$

$(1+cot(\theta))^2 = (csc(\theta))^2$

$1 + 2cot(\theta) + cot^2(\theta) = csc^2(\theta)$

$1 + 2cot(\theta) + cot^2(\theta) = 1 + cot^2(\theta)$ (Pythagorean Identity)

$2cot(\theta) = 0$

$cot(\theta) = 0$

Looking at the unit circle we see that $cot(\theta) = 0$ at $\frac {\pi}{2}$ and $\frac{3\pi}{2}$ . We must plug these back into our original equation to validate them.

Checking $\theta = \frac{\pi}{2}$

$1 + cot(\frac{\pi}{2}) = csc(\frac{\pi}{2})$

Checking $\theta = \frac{3\pi}{2}$

$1 + cot(\frac{3\pi}{2}) = \frac{3\pi}{2}$

And so our only solution is $\theta = \frac{\pi}{2}$

Which of the following is the correct definition of a phase shift?

A measure of the length of a function between vertical asymptotes

The distance a function is shifted diagonally from the general position

The distance a function is shifted horizontally from the general position

The distance a function is shifted vertically from the general position

Explanation

Take the function for example. The graph for is

If we were to change the function to $cos(x + \pi)$ , our phase shift is $-\pi$ . This means we need to shift our entire graph $\pi$ units to the left.

Our new graph $cos(x + \pi)$ is the following

Solve the following equation by squaring both sides: $1 + cot(\theta) = csc(\theta)$

$\theta = 0, \pi$

$\theta = 0$

$\theta = \frac{\pi}{2}$

$\theta = \frac{\pi}{2}, \frac{3\pi}{2}$

Explanation

We begin with our original equation:

$1 + cot(\theta) = csc(\theta)$

$(1+cot(\theta))^2 = (csc(\theta))^2$

$1 + 2cot(\theta) + cot^2(\theta) = csc^2(\theta)$

$1 + 2cot(\theta) + cot^2(\theta) = 1 + cot^2(\theta)$ (Pythagorean Identity)

$2cot(\theta) = 0$

$cot(\theta) = 0$

Looking at the unit circle we see that $cot(\theta) = 0$ at $\frac {\pi}{2}$ and $\frac{3\pi}{2}$ . We must plug these back into our original equation to validate them.

Checking $\theta = \frac{\pi}{2}$

$1 + cot(\frac{\pi}{2}) = csc(\frac{\pi}{2})$

Checking $\theta = \frac{3\pi}{2}$

$1 + cot(\frac{3\pi}{2}) = \frac{3\pi}{2}$

And so our only solution is $\theta = \frac{\pi}{2}$

Which of the following is positive?

$\cos 45$

$\cos 90$

$\cos 135$

$\cos 180$

Explanation

Which of the following is positive?

$\cos 45$

$\cos 90$

$\cos 135$

$\cos 180$

Explanation

Solve the following equation by squaring both sides: $1 + cot(\theta) = csc(\theta)$

$\theta = 0, \pi$

$\theta = 0$

$\theta = \frac{\pi}{2}$

$\theta = \frac{\pi}{2}, \frac{3\pi}{2}$

Explanation

We begin with our original equation:

$1 + cot(\theta) = csc(\theta)$

$(1+cot(\theta))^2 = (csc(\theta))^2$

$1 + 2cot(\theta) + cot^2(\theta) = csc^2(\theta)$

$1 + 2cot(\theta) + cot^2(\theta) = 1 + cot^2(\theta)$ (Pythagorean Identity)

$2cot(\theta) = 0$

$cot(\theta) = 0$

Looking at the unit circle we see that $cot(\theta) = 0$ at $\frac {\pi}{2}$ and $\frac{3\pi}{2}$ . We must plug these back into our original equation to validate them.

Checking $\theta = \frac{\pi}{2}$

$1 + cot(\frac{\pi}{2}) = csc(\frac{\pi}{2})$

Checking $\theta = \frac{3\pi}{2}$

$1 + cot(\frac{3\pi}{2}) = \frac{3\pi}{2}$

And so our only solution is $\theta = \frac{\pi}{2}$

What is the domain of f(x) = sin x?

All positive numbers and 0

All negative numbers and 0

All real numbers except 0

All real numbers

Explanation

The domain of a function is the range of all possible inputs, or x-values, that yield a real value for f(x). Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Sine is defined as the ratio between the side length opposite to the angle in question and the hypotenuse (SOH, or sin x = opposite/hypotenuse). In any triangle created by the angle x and the x-axis, the hypotenuse is a nonzero number. As a result, the denominator of the fraction created by the definition sin x = opposite/hypotenuse is not equal to zero for any angle value x. Therefore, the domain of f(x) = sin x is all real numbers.

What is the domain of f(x) = sin x?

All positive numbers and 0

All negative numbers and 0

All real numbers except 0

All real numbers

Explanation

Which of the following is the correct definition of a phase shift?

A measure of the length of a function between vertical asymptotes

The distance a function is shifted diagonally from the general position

The distance a function is shifted horizontally from the general position

The distance a function is shifted vertically from the general position

Explanation

Take the function for example. The graph for is

If we were to change the function to $cos(x + \pi)$ , our phase shift is $-\pi$ . This means we need to shift our entire graph $\pi$ units to the left.

Our new graph $cos(x + \pi)$ is the following

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