Precalculus Trigonometry Syllabus

In right-triangle applications, what is the difference between an angle of elevation and an angle of depression?

An angle of elevation is measured upward from the horizontal to a line of sight above it; an angle of depression is measured downward from the horizontal to a line of sight below it. They are equal (alternate interior angles) for the same line of sight.

On the unit circle, how are $\cos\theta$ and $\sin\theta$ defined for a point on the circle?

For the point $(x, y)$ where the terminal side of angle $\theta$ meets the unit circle, $\cos\theta = x$ and $\sin\theta = y$.

On the unit circle, how is $\tan\theta$ defined in terms of the coordinates $(x, y)$?

$\tan\theta = \frac{y}{x}$ (with $x \neq 0$), the ratio of the $y$-coordinate to the $x$-coordinate.

What are the coordinates on the unit circle at $\theta = 0$, $\frac{\pi}{2}$, $\pi$, and $\frac{3\pi}{2}$?

$(1,0)$ at $0$, $(0,1)$ at $\frac{\pi}{2}$, $(-1,0)$ at $\pi$, and $(0,-1)$ at $\frac{3\pi}{2}$.

In which quadrants is each trigonometric function positive (the ASTC / 'All Students Take Calculus' rule)?

Quadrant I: all positive; Quadrant II: sine (and cosecant) positive; Quadrant III: tangent (and cotangent) positive; Quadrant IV: cosine (and secant) positive.

In which quadrant is $\sin\theta > 0$ but $\cos\theta < 0$?

Quadrant II, where the $y$-coordinate is positive and the $x$-coordinate is negative.

What is the sign of $\tan\theta$ in each of the four quadrants?

Positive in Quadrants I and III (where $x$ and $y$ have the same sign); negative in Quadrants II and IV (where $x$ and $y$ have opposite signs).

Define a reference angle.

A reference angle is the acute angle (between $0$ and $\frac{\pi}{2}$) formed between the terminal side of $\theta$ and the horizontal $x$-axis.

How do you find the reference angle for an angle $\theta$ in each quadrant (for $0 \le \theta < 2\pi$)?

QI: $\theta$; QII: $\pi - \theta$; QIII: $\theta - \pi$; QIV: $2\pi - \theta$.

How are reference angles used to find exact trig values of non-acute angles?

Evaluate the trig function of the reference angle, then attach the correct sign based on the quadrant of $\theta$ (using the ASTC rule).

What are the amplitude, period, and range of $y = \sin x$ and $y = \cos x$?

Amplitude $= 1$, period $= 2\pi$, and range $= [-1, 1]$ for both.

For $y = a\sin(bx)$ or $y = a\cos(bx)$, what are the amplitude and period?

Amplitude $= |a|$ and period $= \frac{2\pi}{|b|}$.

Compare the graphs of $y = \sin x$ and $y = \cos x$ in terms of a horizontal shift.

The cosine graph is the sine graph shifted left by $\frac{\pi}{2}$: $\cos x = \sin\left(x + \frac{\pi}{2}\right)$. Sine passes through the origin (odd); cosine has a maximum at $x=0$ (even).

What is the period of $y = \tan x$ and where are its vertical asymptotes?

Period $= \pi$; vertical asymptotes occur where $\cos x = 0$, i.e., at $x = \frac{\pi}{2} + \pi k$ for integer $k$.

What is the period of $y = \cot x$ and where are its vertical asymptotes?

Period $= \pi$; vertical asymptotes occur where $\sin x = 0$, i.e., at $x = \pi k$ for integer $k$.

Compare the behavior of $y = \tan x$ and $y = \cot x$ between consecutive asymptotes.

$\tan x$ is increasing on each interval between its asymptotes; $\cot x$ is decreasing on each interval between its asymptotes.

Where are the vertical asymptotes of $y = \sec x$ and what is its range?

Asymptotes where $\cos x = 0$, i.e., $x = \frac{\pi}{2} + \pi k$; range is $(-\infty, -1] \cup [1, \infty)$.

Where are the vertical asymptotes of $y = \csc x$ and what is its range?

Asymptotes where $\sin x = 0$, i.e., $x = \pi k$; range is $(-\infty, -1] \cup [1, \infty)$.

What are the period of $y = \sec x$ and $y = \csc x$?

Both have period $2\pi$, matching their reciprocal functions cosine and sine.

What is the domain and range of $y = \sin^{-1} x$ (arcsine)?

Domain $[-1, 1]$; range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.

What is the domain and range of $y = \cos^{-1} x$ (arccosine)?

Domain $[-1, 1]$; range $[0, \pi]$.

What is the domain and range of $y = \tan^{-1} x$ (arctangent)?

Domain $(-\infty, \infty)$; range $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

Why must the domains of sine, cosine, and tangent be restricted to define their inverses?

The trig functions are periodic and therefore not one-to-one; restricting the domain to an interval where each is one-to-one allows a well-defined inverse that passes the horizontal line test.

On what restricted domain is $\sin x$ made one-to-one to define arcsine?

$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, where sine increases from $-1$ to $1$.

On what restricted domain is $\cos x$ made one-to-one to define arccosine?

$[0, \pi]$, where cosine decreases from $1$ to $-1$.

Evaluate $\sin^{-1}\left(\frac{1}{2}\right)$, $\cos^{-1}(0)$, and $\tan^{-1}(1)$.

$\sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6}$, $\cos^{-1}(0) = \frac{\pi}{2}$, $\tan^{-1}(1) = \frac{\pi}{4}$.

Simplify $\sin\left(\sin^{-1} x\right)$ and state its valid domain.

$\sin\left(\sin^{-1} x\right) = x$ for all $x \in [-1, 1]$.

Does $\sin^{-1}(\sin x) = x$ for all $x$? Explain.

No. $\sin^{-1}(\sin x) = x$ only when $x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. Outside that interval it returns the angle in the range of arcsine that has the same sine value.

Simplify $\cos\left(\sin^{-1} x\right)$ to an algebraic expression.

$\cos\left(\sin^{-1} x\right) = \sqrt{1 - x^{2}}$, using a right triangle with opposite $x$ and hypotenuse $1$.

Simplify $\tan\left(\cos^{-1} x\right)$ to an algebraic expression.

$\tan\left(\cos^{-1} x\right) = \frac{\sqrt{1 - x^{2}}}{x}$, from a right triangle with adjacent $x$ and hypotenuse $1$ (so opposite $= \sqrt{1-x^{2}}$).