Quantitative Reasoning Study Guide for the DAT

Trigonometry

Trigonometry is a branch of mathematics that deals with the study of relationships between the angles and sides of triangles. It encompasses the trigonometric functions sine, cosine, tangent, and their inverses, along with various identities and formulas. Trigonometry plays a crucial role in various fields such as engineering, physics, astronomy, and navigation, where understanding angles and their properties is essential for solving real-world problems.

Right Triangles

A right triangle is a triangle with one angle measuring \(90\) degrees, which is known as a right angle. In right triangles, the ratios of the lengths of the sides to each other depend only on the angles and are independent of the size of the triangle. These ratios define the trigonometric functions, the main ones being sine, cosine, and tangent (abbreviated as sin, cos, and tan, respectively). All of these ratios are based on an angle, \(\theta\). The functions are defined below:

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

The mnemonic SOHCAHTOA can be used to remember these ratios.

Note: There’s also a special trigonometric identity, \(\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}}\), for any angle \(\theta\).

Let’s solve an example problem using the ratios we just learned.

Solve for \(x\) given that \(\cos{B} = \frac{3}{4}\).

Insert at this point in text: 44 Right Triangle-Marked

Using the given information, we can write:

\[\cos{B} = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{15}{x} = \frac{3}{4}\]

Using cross-multiplication, we can solve for \(x\):

\[\frac{15}{x} = \frac{3}{4}\] \[3x = 15 \times 4\] \[3x = 60\] \[x = 20\]

Reciprocal Trigonometric Functions

There are three important trigonometric functions that are the reciprocals of the main functions: cosecant, secant, and cotangent. They are defined as the reciprocals of sine, cosine, and tangent, respectively. Their ratios and abbreviations are as follows:

\[\csc{\theta} = \frac{1}{\sin{\theta}} = \frac{\text{hypotenuse}}{\text{opposite}}\] \[\sec{\theta} = \frac{1}{\cos{\theta}} = \frac{\text{hypotenuse}}{\text{adjacent}}\] \[\cot{\theta} = \frac{1}{\tan{\theta}} = \frac{\text{adjacent}}{\text{opposite}}\]

Degrees and Radians of a Circle

In trigonometry, angles are commonly measured in degrees and radians. A degree is a unit of angular measure equal to \(\frac{1}{360}\) of a full rotation, with a circle containing \(360\) degrees. The symbol for degree is \(\circ\) (e.g., \(360^{\circ}\)).

A radian is a unit of angular measure defined as the angle subtended by an arc of a circle that has the same length as the radius of the circle. In a circle, there are \(2\pi\) radians, which is equivalent to \(360\) degrees. Radians are very common in higher trigonometry (there is no symbol for radians).

The conversion rate between degrees and radians is \(\pi\) radians \(= 180\) degrees. To convert from degrees to radians, multiply the given value by \(\frac{\pi}{180}\). For radians to degrees, multiply by \(\frac{180}{\pi}\).

Convert \(45^{\circ}\) to radians.

\[45^{\circ} \times \frac{\pi \; \text{radians}}{180^{\circ}}\] \[= \frac{45 \pi}{180} \; \text{radians}\] \[= \frac{\pi}{4}\,\text{radians}\]

It is common to leave radian measurement in terms of \(\pi\).

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent can also be evaluated for angles greater than \(90^{\circ}\) using the unit circle. The unit circle is a circle with a radius of \(1\) centered at the origin of a coordinate plane.

To find the values of trigonometric functions for angles greater than \(90^{\circ}\) degrees that are multiples of \(30^{\circ}\), \(45^{\circ}\), and \(60^{\circ}\), we draw the angle in standard position, create a reference triangle within the unit circle, and use the properties of right triangles to find the values of sine, cosine, and tangent for the reference angle. Let’s illustrate this with an example.

Find the value of \(\cos{240^{\circ}}\).

First, we draw the angle \(240^{\circ}\) in standard position in the coordinate plane:

Note that the reference angle is \(60^{\circ}\) past the \(x\)-axis. The triangle formed is a \(30\)-\(60\)-\(90\) triangle. The ratio of its sides are \(1:\sqrt{3}:2\). The value of \(\cos{60^{\circ}}\) is \(0.5\). Since the angle is in the third quadrant, cosine is negative. Thus, \(\cos{240^{\circ}} = -0.5\). Below are some frequently used values of \(\sin\), \(\cos\), and \(\tan\):

Inverse Functions

Inverse trigonometric functions provide a way to find the angle corresponding to a given trigonometric ratio. Three common inverse trigonometric functions are arcsin (or \(\sin{^{-1}}\)), arccos (\(\cos{^{-1}}\)), and arctan (\(\tan{^{-1}}\)). These functions have restricted domains and ranges, which are related to the domains and ranges of the original trigonometric functions.

The arcsin function returns the angle whose sine is a given value. Its domain is \([-1,1]\) since the sin function’s range is \([-1,1]\). The range of arcsin is \([-90^{\circ}, 90^{\circ}]\) or \(\left[\frac{\pi}{2}, \frac{\pi}{2}\right]\) radians.

The arccos function returns the angle whose cosine is a given value. Its domain is the same as arcsin’s, but its range is \([0^{\circ}, 180^{\circ}]\) or \([0, \pi]\) radians.

The arctan function returns the angle whose tangent is a given value. Its domain is all real numbers, and its range is \([-90^{\circ}, 90^{\circ}]\) or \(\left[\frac{\pi}{2}, \frac{\pi}{2}\right]\) radians.

Find the angle whose sine is \(\frac{1}{2}\).

Given \(\sin{\theta} = \frac{1}{2}\), we have \(\theta = \arcsin\left(\frac{1}{2} \right)\). To find this angle, let’s draw a right triangle. We know that the sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, the side opposite the angle is \(1\) (since sine is opposite over hypotenuse), and the hypotenuse is \(2\) (since the sine is given as \(\frac{1}{2}\)).

Now, we can use the Pythagorean theorem to find the length of the third side of the triangle (the side adjacent to the angle):

\[1^2 + x^2 = 2^2\] \[x^2 = 3\] \[x = \sqrt{3}\]

Therefore, the angle \(\theta\) corresponding to the given sine ratio of \(\frac{1}{2}\) is the angle whose opposite side has length \(1\) and adjacent side has length \(\sqrt{3}\). That is, the ratio of the sides of the right triangle formed is \(1:\sqrt{3}:2\), so we know we are dealing with a \(30-60-90\) right triangle. Since the side opposite to \(\theta\) has the smallest side length, then we can conclude that \(\theta\) is the smallest of the three angles in the triangle. Therefore, \(\theta = 30^{\circ}\).

For more complicated problems, it might be easier to use a calculator. Using the same example as above, given \(\sin{\theta} = \frac{1}{2}\), we have \(\theta = \arcsin\left(\frac{1}{2} \right)\). We can simply enter \(\arcsin\left(\frac{1}{2} \right)\) into the calculator to find that \(\theta = 30^{\circ} \text{ or } \frac{\pi}{6} \, \text{radians}\).

Mathematical Reasoning

There are several types of questions in the Quantitative Reasoning section of the DAT that require intensive reasoning to find the answer. Read below to learn more about them.

Quantitative Comparison Questions:

This type of question will present information about two quantities and ask you which is greater, if they are equal, or if you cannot determine the comparison of quantity with only the information given in the question. Here are two examples:

Example 1

\(a = 2b\), \(b\) is a positive integer.

\[\underline{\text{Quantity A}} \hspace{1.5cm} \underline{\text{Quantity B}}\] \[9^{2b} \hspace{3.2cm} 3^{a}\]

A. Quantity A is greater.

B. Quantity B is greater.

C. The two quantities are equal.

D. The relationship cannot be determined from the information given.

Solution

Replacing \(a= 2b\) in the second quality yields \(3^{2b}\). Now we are comparing \(9^{2b}\) with \(3^{2b}\). Since both powers are the same, it’s clear \(9^{2b}\) would be greater than \(3^{2b}\). The correct answer is A.

Example 2

A set consists of \(15\) consecutive integers.

A. Quantity A is greater.

B. Quantity B is greater.

C. The two quantities are equal.

D. The relationship cannot be determined from the information given.

Solution

In a set of \(15\) consecutive integers, the middle value would be the median. If the first number is odd, then there would be \(4\) odd and \(3\) even numbers both below and above the median. If the first number is even, then the results would be reversed.

In the first case, the probability of selecting an odd number greater than the median would be \(\frac{4}{15}\), and the probability of selecting an ever number less than the median would be \(\frac{3}{15}\). Quantity A would be greater. In the reverse case, the probabilities will also be reserved, so Quantity B would be greater. Since more than one relationship is possible, the correct answer is D, it can’t be determined from the information given.

Data Sufficiency Questions:

These questions will begin with a question and two numbered statements, 1 and 2. You will then be asked to determine if either statement alone is enough to provide a singular answer for the question, if it takes both of them to support an answer, or if an answer cannot be determined even with the information given in both statements. Let’s look at a couple examples:

Example 1

What is the value of \(t\)?

1. \[t^2 = 36\]
2. \[2t(t+6) = 0\]

A) Statement 1 ALONE is sufficient, but statement 2 ALONE is not sufficient.
B) Statement 2 ALONE is sufficient, but statement 1 ALONE is not sufficient.
C) Both statements TOGETHER are sufficient, but neither statement ALONE is sufficient.
D) Each statement ALONE is sufficient.
E) Statements 1 and 2 TOGETHER are NOT sufficient.

Solution

If we consider statement 1 alone, we get two values of \(t\):

\[t = \sqrt{36}\] \[t = -6, 6\]

If we consider statement 2 alone, we again get two values of \(t\):

\[2t(t+6) = 0\] \[t = 0, -6\]

If we consider both statements together, there is one distinct value of \(t\) that satisfies both conditions, which is \(t = -6\). So, only combining both statements leads to sufficiency. The correct answer is C.

Example 2

What is the value of the integer \(x\)?

1. \[15 \leq x \leq 22\]
2. \(x\) is a prime number.

A) Statement 1 ALONE is sufficient, but statement 2 ALONE is not sufficient
B) Statement 2 ALONE is sufficient, but statement 1 ALONE is not sufficient
C) Both statements TOGETHER are sufficient, but neither statement ALONE is sufficient
D) Each statement ALONE is sufficient
E) Statements 1 and 2 TOGETHER are NOT sufficient

Solution

If we consider statement 1 alone, we don’t have a specific answer since there are a range of values of \(x\). If we consider statement 2 alone, we have an infinite choice since there are infinite primes. If we consider both statements together, there are two primes (\(17\) and \(19\)) between the range \(15 \leq x \leq 22\). So, the correct answer is E, we cannot determine an answer based on these two statements alone.

Data Interpretation Questions

These mathematical reasoning questions are based on one or more graphics, such as charts, tables, figures, graphs, or images. If there is more than one graphic, you will need to refer to all that are provided. If the resource is a graph, be sure to look at all parts of it: the axis, labels, title, data, key, and scale. There may also be added notes for reference.

There may be more than enough data in the graphic than that needed to answer the question, so be sure you know exactly what the question is asking. Don’t be thrown off by complicated-looking graphics. The answer is sometimes easy if you concentrate on your goal.

Here are two examples:

Example 1

For the graphic above shown, in \(2021\), the average annual temperature of City B was approximately what percent of that of City A?

Solution

First, you read the values of the average annual temperatures of the two cities in \(2021\). For City A it is \(53^{\circ}\)F, and for City B it is \(47^{\circ}\)F. To find the percentage of City B to City A, we divide the temperature of City B by City A and then multiply the result by \(100\):

\[\frac{47}{53} \times 100 \approx 88.68\%\]

Example 2

For the first year in which sales from the clothing sector surpassed \(\$60\text{,}000\), approximately what were the total profits of the clothing sector?

A. \(\$8\text{,}000\)
B. \(\$14\text{,}000\)
C. \(\$12\text{,}000\)
D. \(\$7\text{,}000\)

Solution

We have to look at the first chart to determine the sales. Note that the clothing sales are the top parts of the bars of the stacked column chart. So, you have to be extra careful when determining the sales. Looking closely, we can see that in \(2014\), the sales for the clothing sector surpassed \(\$60\text{,}000\). To find the total profits for that year, we look at the second chart, where the clothing profits are the top parts of the bars. From the second chart, you can make an educated guess that the profit for \(2014\) was a bit over \(\$10\text{,}000\). From the answer choices, C is the most reasonable.