Quantitative Reasoning Study Guide for the CLT

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles, particularly right triangles. It is essential in various fields such as physics, engineering, and architecture.

Trigonometric Ratios

Trigonometric ratios are the relationships between the sides of a right triangle relative to one of its angles. The three primary trigonometric ratios are sine, cosine, and tangent. Each ratio is defined as follows:

sine (sin)—the ratio of the length of the opposite side to the hypotenuse

\[\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\]

cosine (cos)—the ratio of the length of the adjacent side to the hypotenuse

\[\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\]

tangent (tan)—the ratio of the length of the opposite side to the adjacent side

\[\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\]

Additionally, we have the reciprocal trigonometric functions (and corresponding ratios as they relate to right triangles), which are defined as follows:

cosecant (csc)—the reciprocal of sine

\[\csc \theta = \frac{1}{\sin \theta} = \frac{\text{hypotenuse}}{\text{opposite}}\]

secant (sec)—the reciprocal of cosine

\[\sec \theta = \frac{1}{\cos \theta} = \frac{\text{hypotenuse}}{\text{adjacent}}\]

cotangent (cot)—the reciprocal of tangent

\[\cot \theta = \frac{1}{\tan \theta} = \frac{\text{adjacent}}{\text{opposite}}\]

Consider a right triangle with an angle $\theta$, where the opposite side is $3$ units, the adjacent side is $4$ units, and the hypotenuse is $5$ units.

12 Right Triangle.png

Using the definition of all the trigonometric ratios, we can calculate the values of each or $\theta$:

\[\sin \theta = \frac{3}{5}\] \[\cos \theta = \frac{4}{5}\] \[\tan \theta = \frac{3}{4}\] \[\csc \theta = \frac{5}{3}\] \[\sec \theta = \frac{5}{4}\] \[\cot \theta = \frac{4}{3}\]

Inverse Trigonometric Functions

For the trigonometric functions sine, cosine, and tangent (and their corresponding reciprocal functions), we input an angle, $\theta$, and the output is a number, $\alpha$. For inverse trigonometric functions, we input the number, say $\alpha$, and the output is the corresponding angle. There are two common ways to write each inverse trigonometric function. The inverse trigonometric functions (and the two ways to write them) are:

\[\arccos \alpha = \cos^{-1}(\alpha)\] \[\arcsin \alpha = \sin^{-1}(\alpha)\] \[\arctan \alpha = \tan^{-1}(\alpha)\]

It is important to note that for arccosine and arcsine, the only valid inputs are for $-1 \leq \alpha \leq 1$. These functions are inverses of their respective trigonometric functions. In other words, if $\cos \theta = b$, then $\arccos b = \theta$. The same idea works for the other two inverse trigonometric functions. It is often helpful to write an angle in the form of an inverse trigonometric function if we can’t quite figure it out.

For example, suppose we are given a right triangle with side lengths $3$ inches and $4$ inches and we want to know the measure of the angle (let’s call it $\theta$) that is opposite the side of length $4$. It would be hard to compute $\theta$ exactly based on the given information. However, since we are dealing with a right triangle, we know:

\[\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3}\]

Therefore, we can conclude that $\theta = \arctan \frac{4}{3} = \tan^{-1}\left( \frac{4}{3} \right)$.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable within a certain range. They help simplify expressions and solve trigonometric equations. These are some of the fundamental identities:

Pythagorean identities—$\sin^2 \theta + \cos^2 \theta = 1, \,1 + \tan^2 \theta = \sec^2 \theta$
quotient identity—$\tan \theta = \frac{\sin \theta}{\cos \theta}$
reciprocal identities—$\csc \theta = \frac{1}{\sin \theta},\, \sec \theta = \frac{1}{\cos \theta}, \,\cot \theta = \frac{1}{\tan \theta}$
double angle identities—$\sin (2 \theta) = 2 \sin \theta \cos \theta, \,\cos (2 \theta) = \cos^2 \theta - \sin^2 \theta$

Let’s look at an example in which we simplify a trigonometric expression using identities.

Rewrite the expression $\frac{\sin \theta}{\cos \theta} + \cot \theta$ using only one trigonometric function.

Solution

First, we can use the identity $\frac{\sin \theta}{\cos \theta} = \tan \theta$ to substitute into the equation:

\[\frac{\sin \theta}{\cos \theta} + \cot \theta\] \[= \tan \theta + \cot \theta\]

Now, we also know the identity $\cot \theta = \frac{1}{\tan \theta}$, which means the expression becomes:

\[= \tan \theta + \frac{1}{\tan \theta}\]

Trigonometric Expressions

Trigonometric expressions involve the use of trigonometric functions such as sine, cosine, tangent, and their reciprocal functions, among others. These functions are fundamental in understanding the relationships between the angles and sides of a triangle, especially in right-angled triangles.

A common task when working with trigonometric expressions is to simplify them by recognizing and applying trigonometric identities. These identities are established relationships between trigonometric functions that allow us to express one function in terms of another. For example, recall one of the Pythagorean identities:

\[\sin^2 \theta + \cos^2 \theta = 1\]

Using this identity, you can recognize that $\sin^2 \theta$ can be rewritten as $1 - \cos^2 \theta$, since rearranging the Pythagorean identity gives $\sin^2 \theta = 1 - \cos^2 \theta$. This demonstrates how two different-looking expressions can actually be identical.

In more complex problems, you may encounter expressions like $\tan^2 \theta$, which can be rewritten using the definition of tangent as $\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}$ or recognized as $\sec^2 \theta - 1$ using the other Pythagorean identity ($1 + \tan^2 \theta = \sec^2 \theta$).

Being able to identify and manipulate these equivalent forms is crucial in simplifying trigonometric expressions, solving trigonometric equations, or proving identities. Whether you’re simplifying an expression or solving an equation, familiarity with these relationships helps reduce complex trigonometric expressions into more manageable forms. For instance, in solving trigonometric equations, using identities can help transform all terms into a single trigonometric function, making the equation easier to handle.

Graphs of Trigonometric Functions

The graphs of trigonometric functions, such as $\sin x$, $\cos x$, and $\tan x$, provide a visual understanding of their behavior over different intervals on the $x$-axis.

sine graph ($y = \sin x$)

13 Sine Graph.png

The sine wave starts at $0$, reaches a maximum of $1$ at $\frac{\pi}{2}$, returns to $0$ at $\pi$, reaches a minimum of $-1$ at $\frac{3\pi}{2}$, and repeats.

cosine graph ($y = \cos x$)

14 Cosine Graph.png

The cosine wave starts at $1$, decreases to $0$ at $\frac{\pi}{2}$, reaches a minimum of $-1$ at $\pi$, and returns to $1$ at $2\pi$.

tangent graph ($y = \tan x$)

15 Tangent Graph.png

The tangent graph has vertical asymptotes (lines that the graph approaches but never touches, where the function tends toward infinity) at $\frac{\pi}{2}$, and repeats every $\pi$. It increases rapidly, crossing $0$ at multiples of $\pi$.

These are the key characteristics in a trigonometric graph:

amplitude—This is the height of the graph from the centerline to the peak. For a function of the form $a \sin x$ or $a \cos x$, the amplitude is $$

$$.

period—This is the distance between repeating sections of the graph. For a function of the form $\sin (kx)$ or $\cos (kx)$, the period is $$ \frac{2\pi}{

} $. For a function of the form$\tan (kx) $, the period is$\frac{\pi}{

}$$.

symmetry—Sine (and cosecant, the reciprocal of sine) is an odd function, meaning $\sin(-\theta) = -\sin \theta$, while cosine (and secant, the reciprocal of cosine) is an even function, meaning $\cos(-\theta) = \cos \theta$.
phase shift—This is the horizontal shift of the graph.

16 Phase Shift Graph.png

The Unit Circle

The unit circle is a circle that is centered at the origin of a coordinate plane with a radius of $1$. It allows us to define the trigonometric functions for all angles, not just those in right triangles. The coordinates of points on the unit circle correspond to the $x$ and $y$ values of $\cos \theta$ and $\sin \theta$ for a given angle $\theta$.

17 Unit Circle.png

Degrees vs. Radians

A degree is a unit of angular measure in which a full circle is $360^{\circ}$. A radian is another unit of angular measure where a full circle is $2\pi$ radians.

To convert from degrees to radians ($\text{rad}$):

\[\text{rad} = \frac{\pi}{180} \times x^{\circ}\]

To convert from radians to degrees:

\[x^{\circ} = \frac{180}{\pi} \times \text{rad}\]

To convert $90^{\circ}$ to radians:

\[\frac{\pi}{180} \times 90 = \frac{\pi}{2}\]

To convert $\frac{\pi}{3}$ radians to degrees:

\[\frac{180}{\pi} \times \frac{\pi}{3} = 60^{\circ}\]

Angles

Angles on the unit circle are measured counterclockwise from the positive $\boldsymbol{x}$-axis. The side of the angle on the $x$-axis is called the initial side and the other side of the angle is called the terminal side. The coordinates of points on the unit circle for standard angles (e.g., $0^{\circ}, \, 90^{\circ}, \, 180^{\circ},$ and $270^{\circ}$) help determine the values of sine and cosine.

In trigonometry, the unit circle is a tool for analyzing angles and understanding the behavior of trigonometric functions. Each point on the circle corresponds to a specific angle, with the angle’s vertex at the origin $(0, 0)$. The $x$-coordinate of the point on the unit circle represents the cosine of the angle, while the $y$-coordinate represents the sine.

To analyze an angle given on the unit circle, follow these steps:

1) Determine the angle’s position.

Angles on the unit circle are measured from the positive $x$-axis. Counterclockwise rotation produces positive angles, and clockwise rotation gives negative angles. For example, $90^{\circ}$ or $\frac{\pi}{2}$ radians is positioned at the top of the unit circle, while $270^{\circ}$ or $\frac{3\pi}{2}$ radians is at the bottom.

2) Use coordinates.

The coordinates of a point on the unit circle are key to determining the trigonometric values for the angle. For an angle $\theta$, the coordinates $(x, y)$ represent $(\cos \theta, \sin \theta)$. These values give the cosine and sine of the angle, respectively.

For example, $0^{\circ}$ corresponds to the point $(1, 0)$, so $\cos(0^{\circ}) = 1$ and $\sin(0^{\circ}) = 0$. Similarly, $180^{\circ}$ corresponds to $(-1, 0)$, meaning $\cos(180^{\circ}) = -1$ and $\sin(180^{\circ}) = 0$.

3) Consider Quadrants.

The unit circle is divided into four quadrants:

quadrant I—Both $\cos \theta$ and $\sin \theta$ are positive.
quadrant II—$\cos \theta$ is negative, but $\sin \theta$ is positive.
quadrant III—Both $\cos \theta$ and $\sin \theta$ are negative.
quadrant IV—$\cos \theta$ is positive, but $\sin \theta$ is negative.

This helps in quickly identifying the signs of the sine and cosine values for angles in different quadrants.

4) Use reference angles.

Every angle on the unit circle has a corresponding reference angle, which is the acute angle formed by the terminal side of the given angle and the $x$-axis. Reference angles are useful for simplifying the calculation of trigonometric functions. For example, an angle of $135^{\circ}$ has a reference angle of $45^{\circ}$, meaning that $\sin(135^{\circ}) = \sin(45^{\circ})$ but with the appropriate sign based on the quadrant.

18 Reference Angle of Circle.png

5) Use periodicity.

Trigonometric functions are periodic, meaning their values repeat at regular intervals. For sine and cosine, the period is $360^{\circ}$ (or $2\pi$ radians), meaning that $\sin \theta = \sin(\theta + 360^{\circ})$ and $\cos \theta = \cos(\theta + 360^{\circ})$.

By using the unit circle, you can efficiently analyze any angle, understand the trigonometric functions associated with it, and solve problems involving angles and their corresponding values.

Values Along the Unit Circle

sine and cosine—Recall that on the unit circle, any point $(x, y)$ corresponding to an angle $\theta$ has coordinates $(\cos \theta, \sin \theta)$. This means that the $x$-coordinate gives the cosine of the angle, and the y-coordinate gives the sine.

For example, at $30^{\circ}$ (or $\frac{\pi}{6}$ radians), the point is $\left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right)$, so $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$ and $\sin 30^{\circ} = \frac{1}{2}$.

At $60^{\circ}$ (or $\frac{\pi}{3}$ radians), the point is $\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)$, so $\cos 60^{\circ} = \frac{1}{2}$ and $\sin 60^{\circ} = \frac{\sqrt{3}}{2}$.

tangent—The tangent of an angle $\theta$ is defined as the ratio of the sine to the cosine:

\[\tan \theta = \frac{\sin \theta}{\cos \theta}\]

For angles where $\cos \theta = 0$, the tangent is undefined. For example, at $90^{\circ}$, the cosine is $0$, so $\tan 90^{\circ}$ is undefined. The tangent function is periodic with a period of $180^{\circ}$ or $\pi$ radians.

cosecant, secant, and cotangent—The lesser-used trigonometric functions are cosecant, secant, and cotangent, which are the reciprocals of sine, cosine, and tangent, respectively:

\[\csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \cot \theta = \frac{1}{\tan \theta}\]

These functions are undefined where their respective trigonometric functions equal $0$. For example, $\sec 90^{\circ}$ is undefined because $\cos 90^{\circ} = 0$.

By using the unit circle and understanding these properties, you can compute trigonometric function values for any angle and use them to solve real-world problems involving angles, distances, and periodic phenomena.