Precalculus Study Notes: Angles, Trigonometric Functions, and Their Graphs

Angles and Their Measure

Degree and Radian Measure

Angles can be measured in degrees or radians. Radian measure is directly related to the arc length of a circle, making it a natural unit in mathematics.

• Degree: A full rotation is 360°.

• Radian: A full rotation is $2\pi$ radians.

• Conversion:

  • Degrees to radians: $\theta_{\text{rad}} = \theta_{\text{deg}} \cdot \frac{\pi}{180}$

  • Radians to degrees: $\theta_{\text{deg}} = \theta_{\text{rad}} \cdot \frac{180}{\pi}$

• Example: Convert $40^\circ$ to radians: $40 \cdot \frac{\pi}{180} = \frac{2\pi}{9}$ radians.

• Example: Convert $\frac{11\pi}{6}$ radians to degrees: $\frac{11\pi}{6} \cdot \frac{180}{\pi} = 330^\circ$.

Degrees, Minutes, and Seconds

For greater precision, each degree is divided into 60 minutes (′) and each minute into 60 seconds (″).

• Conversion to Decimal Degrees: $D^\circ M' S'' = D + \frac{M}{60} + \frac{S}{3600}$

• Example: $70^\circ 18' 13'' = 70 + \frac{18}{60} + \frac{13}{3600} \approx 70.3036^\circ$

• Conversion from Decimal Degrees: $110.433^\circ = 110^\circ 25' 59''$

Arc Length and Area of a Sector

The arc length and area of a sector depend on the angle (in radians) and the radius of the circle.

• Arc Length: $s = r\theta$ (with $\theta$ in radians)

• Arc Length (degrees): $s = r\theta \cdot \frac{\pi}{180}$

• Area of Sector: $A = \frac{1}{2} r^2 \theta$ (with $\theta$ in radians)

• Area (degrees): $A = \frac{1}{2} r^2 \theta \cdot \frac{\pi}{180}$

• Example: For $r = 1.5$ ft, $\theta = 57^\circ$:

  • Convert $57^\circ$ to radians: $57 \cdot \frac{\pi}{180} \approx 0.995$ radians

  • Arc length: $s = 1.5 \times 0.995 \approx 1.49$ ft

  • Area: $A = \frac{1}{2} \times 1.5^2 \times 0.995 \approx 1.12$ ft2

Applications: Distances on the Earth

To find the distance between two points on the Earth's surface (assuming a spherical Earth):

• Find the difference in latitude in degrees, convert to radians.

• Use $s = r\theta$ with $r = 3960$ miles (Earth's radius).

• Example: Pittsburgh ($40^\circ 30'$ N) and Charlotte ($32^\circ 54'$ N):

  • Difference: $7^\circ 36' = 7.6^\circ$

  • Radians: $7.6 \cdot \frac{\pi}{180} \approx 0.133$

  • Distance: $3960 \times 0.133 \approx 527$ miles

The Unit Circle and Trigonometric Functions

The Unit Circle

The unit circle is a circle of radius 1 centered at the origin. It is fundamental in defining trigonometric functions for all real numbers.

• Equation: $x^2 + y^2 = 1$

• Key Angles: $0^\circ$ ($0$), $90^\circ$ ($\frac{\pi}{2}$), $180^\circ$ ($\pi$), $270^\circ$ ($\frac{3\pi}{2}$), $360^\circ$ ($2\pi$)

• Example: Is $\left(\frac{13}{7}, \frac{6}{7}\right)$ on the unit circle? $\left(\frac{13}{7}\right)^2 + \left(\frac{6}{7}\right)^2 = \frac{169+36}{49} = \frac{205}{49} \neq 1$ (so, not on the unit circle).

Primary Trigonometric Functions

For a point $(x, y)$ on the unit circle corresponding to angle $\theta$:

• $\cos \theta = x$

• $\sin \theta = y$

• $\tan \theta = \frac{y}{x}$ (if $x \neq 0$)

• Example: For $\theta = 30^\circ$ ($\frac{\pi}{6}$), $\cos 30^\circ = \frac{\sqrt{3}}{2}$, $\sin 30^\circ = \frac{1}{2}$

Special Triangles and Trigonometric Values

• Equilateral Triangle (30°, 60°, 90°): Splitting an equilateral triangle of side 1 gives a $30$-$60$-$90$triangle with sides $1$, $\frac{1}{2}$, $\frac{\sqrt{3}}{2}$.

• Isosceles Right Triangle (45°, 45°, 90°): Sides $1$, $1$, $\sqrt{2}$.

• Key Values:

  • $\sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}$

  • $\sin 60^\circ = \frac{\sqrt{3}}{2}$, $\cos 60^\circ = \frac{1}{2}$

  • $\sin 30^\circ = \frac{1}{2}$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$

Secondary Trigonometric Functions

These are defined as reciprocals of the primary functions:

• $\sec \theta = \frac{1}{\cos \theta}$

• $\csc \theta = \frac{1}{\sin \theta}$

• $\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$

• Example: For $(x, y) = \left(\frac{2}{7}, \frac{3\sqrt{5}}{7}\right)$:

  • $\cos \theta = \frac{2}{7}$

  • $\sin \theta = \frac{3\sqrt{5}}{7}$

  • $\tan \theta = \frac{3\sqrt{5}}{2}$

  • $\sec \theta = \frac{7}{2}$

  • $\csc \theta = \frac{7}{3\sqrt{5}}$

  • $\cot \theta = \frac{2}{3\sqrt{5}}$

Evaluating Trigonometric Functions on the Unit Circle

• Example: $\sin\left(\frac{\pi}{2}\right) = 1$

• Example: $\cot\left(\frac{3\pi}{4}\right) = -1$

• Example: $\cos\left(\frac{5\pi}{4}\right) = -\frac{1}{\sqrt{2}}$

Applications of Trigonometric Functions

Trigonometric functions model periodic phenomena, such as population cycles.

• Example: $N(t) = 450 \sin\left(\frac{\pi t}{6}\right) + 1550$ models deer population, $t$ years after 2010.

• Find $N(2)$ (2012): $N(2) = 450 \sin\left(\frac{\pi}{3}\right) + 1550 = 450 \cdot \frac{\sqrt{3}}{2} + 1550 \approx 1948$

Trigonometric Functions of Angles

SOH-CAH-TOA and Right Triangles

Trigonometric functions can be defined using right triangles:

• $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$

• $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

• $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$

• $\csc \theta = \frac{\text{Hypotenuse}}{\text{Opposite}}$

• $\sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent}}$

• $\cot \theta = \frac{\text{Adjacent}}{\text{Opposite}}$

Trigonometric Functions for Arbitrary Points

For a point $(x, y)$ (not necessarily on the unit circle), with $r = \sqrt{x^2 + y^2}$:

• $\sin \theta = \frac{y}{r}$

• $\cos \theta = \frac{x}{r}$

• $\tan \theta = \frac{y}{x}$

• $\csc \theta = \frac{r}{y}$

• $\sec \theta = \frac{r}{x}$

• $\cot \theta = \frac{x}{y}$

• Example: For $(7, -2)$, $r = \sqrt{7^2 + (-2)^2} = \sqrt{49 + 4} = \sqrt{53}$

  • $\sin \theta = \frac{-2}{\sqrt{53}}$

  • $\cos \theta = \frac{7}{\sqrt{53}}$

  • $\tan \theta = \frac{-2}{7}$

  • $\csc \theta = -\frac{\sqrt{53}}{2}$

  • $\sec \theta = \frac{\sqrt{53}}{7}$

  • $\cot \theta = -\frac{7}{2}$

Pythagorean Identities

These identities are fundamental relationships among trigonometric functions:

• $\sin^2 \theta + \cos^2 \theta = 1$

• $\tan^2 \theta + 1 = \sec^2 \theta$

• $1 + \cot^2 \theta = \csc^2 \theta$

Reference Angles

The reference angle $\theta_R$ is the acute angle between the terminal side of $\theta$ and the x-axis.

• First Quadrant: $\theta_R = \theta$

• Second Quadrant: $\theta_R = 180^\circ - \theta$

• Third Quadrant: $\theta_R = \theta - 180^\circ$

• Fourth Quadrant: $\theta_R = 360^\circ - \theta$

• Example: Reference angle for $500^\circ$: $500^\circ - 360^\circ = 140^\circ$; $180^\circ - 140^\circ = 40^\circ$ (since $140^\circ$ is in the second quadrant, reference angle is $40^\circ$).

Graphs of Trigonometric Functions

Graphs of Sine and Cosine

The graphs of $y = \sin x$ and $y = \cos x$ are periodic and have characteristic shapes.

• Amplitude: $|a|$ in $y = a \sin x$ or $y = a \cos x$ (vertical stretch)

• Period: $\frac{2\pi}{|b|}$ in $y = a \sin(bx)$ or $y = a \cos(bx)$ (horizontal stretch/shrink)

• Phase Shift: $-\frac{c}{b}$ in $y = a \sin(bx + c)$ or $y = a \cos(bx + c)$ (horizontal shift)

• Example: $y = 3 \sin(4x)$ has amplitude 3, period $\frac{2\pi}{4} = \frac{\pi}{2}$

• Example: $y = -4 \cos(7x)$ has amplitude 4, period $\frac{2\pi}{7}$

Phase Shift

• In $y = a \sin(bx + c)$, phase shift is $-\frac{c}{b}$

• Example: $y = 3 \sin(4x + \pi)$ has phase shift $-\frac{\pi}{4}$

Graphing Tips

• Identify amplitude, period, and phase shift.

• Mark key points: zeros, maxima, minima.

• Sketch one period, then extend as needed.

Graphs of Tangent and Cotangent

The graphs of $y = \tan x$ and $y = \cot x$ have vertical asymptotes and repeat every $\pi$ units.

• Period: $\pi$ for both $\tan x$ and $\cot x$

• Vertical Asymptotes:

  • $y = \tan x$: at $x = \frac{\pi}{2} + k\pi$

  • $y = \cot x$: at $x = k\pi$

• Example: $y = 2 \tan(3x + \frac{\pi}{2})$ has period $\frac{\pi}{3}$ and phase shift $-\frac{\pi}{6}$

Graphs of Secant and Cosecant

Secant and cosecant are the reciprocals of cosine and sine, respectively. Their graphs have vertical asymptotes where the denominator is zero.

• $y = \sec x$ is undefined where $\cos x = 0$

• $y = \csc x$ is undefined where $\sin x = 0$

• Graph by plotting $y = \cos x$ or $y = \sin x$ and drawing the reciprocal curves, with asymptotes at zeros of the base function.

Tangent as a Slope

The tangent of an angle gives the slope of a line making that angle with the positive x-axis.

• Equation of a line: $y - y_1 = m(x - x_1)$, where $m = \tan \theta$

• Example: Through $(2, 5)$, angle $60^\circ$: $m = \tan 60^\circ = \sqrt{3}$, so $y - 5 = \sqrt{3}(x - 2)$

Table: Summary of Trigonometric Functions

Function	Definition (Unit Circle)	Definition (Right Triangle)	Reciprocal
sin θ	y	Opp/Hyp	csc θ
cos θ	x	Adj/Hyp	sec θ
tan θ	y/x	Opp/Adj	cot θ
csc θ	1/y	Hyp/Opp	sin θ
sec θ	1/x	Hyp/Adj	cos θ
cot θ	x/y	Adj/Opp	tan θ

Additional info:

• Some examples and values were inferred for clarity and completeness.

• Graphing instructions are summarized for practical application.