Math Study Guide for the SAT Exam

Geometry: Circles

A circle is a unique type of shape in that it is not composed of straight lines. Instead, it is a curve created by a collection of points equidistant from a point at its center. Recall that the full span of a circle is \(360\) degrees.

Parts of a Circle

Any line segment starting and ending on the circle that passes through the center is known as a diameter. A radius is any line segment starting at the circle’s center and ending on the circle. Line segments that do not pass through the center of the circle are called chords.

The circumference of a circle is the perimeter, or the distance around the circle. It is defined as:

\[C = \pi d\quad\text{ or } \quad C = 2 \pi r\]

where \(d\) is the diameter and \(r\) is the radius.

An arc is a part of the circumference designated by two or three points. It can be represented by \(\stackrel{\Large\frown}{CH}\) (this means arc \(CH\)). Arc length (\(s\)) is defined mathematically in radians, which is a unit of angular measure based on the radius of the circle. Specifically, a radian is the angle formed when the length of the arc is equal to the radius of the circle. This is the formula for arc length: \(s = r \theta\)

where \(r\) is the radius and \(\theta\) is the measurement of the central angle. It can also be defined in degrees with this formula:

\[s = r \theta \frac{\pi}{180}\]

A sector is the area enclosed by a central angle. It is defined mathematically in radians by this formula:

\[A = \frac{\theta}{2} r^2\]

where \(\theta\) is the central angle measurement and \(r\) is the radius. The same area is represented in degrees through the following formula:

\[A = \frac{\theta}{360} \pi r^2\]

A central angle is any angle with its side lengths formed from two radii and an arc. An inscribed angle is any angle formed from two chords starting at the same point and an arc.

A tangent line is a line forming a \(90\)-degree angle with a radius of the circle at the point along the circle’s edge.

Trigonometry

Some knowledge of right triangle trigonometry and radian measurements will be necessary for some of the SAT Math section. Refresh your memory by reviewing these concepts. You will not need to find the value of trigonometric functions that require the use of a calculator.

Right Triangles

Right triangles are at the heart of trigonometry. A right triangle is a triangle that has one angle equal to \(90\) degrees, which is often marked with a small square in the corner of the triangle. This right angle distinguishes the hypotenuse (the side opposite the right angle) from the two other sides, called the legs.

Right triangles can vary in shape and size, but every right triangle has the same essential components. There is always one right angle, two acute angles, a hypotenuse, and two legs. Because the sum of the angles in any triangle is always \(180\) degrees, knowing one acute angle allows you to find the other. Right triangles form the basis for many real-world applications of trigonometry, from measuring the heights of objects to analyzing slopes.

All of these are right triangles:

Basic Trigonometric Functions

The trigonometric functions relate the side lengths of a right triangle to its angles. The three most important trig functions are sine, cosine, and tangent. These functions are defined as follows:

\[\sin \theta = \frac{o}{h}\] \[\cos \theta = \frac{a}{h}\] \[\tan \theta = \frac{o}{a}\]

where \(\theta\) is one of the acute angles, \(o\) is the side opposite the angle \(\theta\), \(a\) is the side adjacent to the angle \(\theta\), and \(h\) is the hypotenuse.

You can remember this with the acronym SOHCAHTOA, which is actually three acronyms together. SOH tells you that “sine equals opposite over hypotenuse,” CAH means “cosine equals adjacent over hypotenuse,” and TOA means “tangent equals opposite over adjacent.”

Given an angle (other than the right angle) and any side length, every angle and length of a right triangle can be found using the appropriate trigonometric function.

Let’s do an example.

In the right triangle shown below, angle \(\theta\) measures \(35^{\circ}\). The side adjacent to \(\theta\) has a length of \(12\) units. What is the length of the hypotenuse?

Solution

Since we know the side adjacent to the angle given and want to find the hypotenuse, we will use the trigonometric function \(\cos{}\) since \(\cos \theta = \frac{a}{h}\). Setting up the problem (using \(h\) as the hypotenuse):

\[\cos{35^{\circ}} = \frac{12}{h}\]

Now, we can cross-multiply and solve for \(h\):

\[h \times \cos{35^{\circ}} = 12\] \[h = \frac{12}{\cos{35^{\circ}}}\] \[h \approx 14.65\]

You have a right triangle and know that one of the angles (other than the right angle) is \(\theta = 30^{\circ}\) and the side opposite of this angle is \(3\) inches. What is the length of the hypotenuse of the triangle?

Solution

We know the angle measure and the length of the side opposite of the angle measure. We want to solve for the length of the hypotenuse. The trigonometric function that puts all of this information together is sine. That is:

\[\sin(\theta) = \frac{o}{h} = \frac{3}{h}\]

Also, we know that \(\sin(\theta) = \sin(30^{\circ}) = 0.5\).

Using this information, we can solve for the length of the hypotenuse:

\[\sin(\theta) = \sin(30^{\circ}) = 0.5 = \frac{3}{h}\]

Thus, the length of the hypotenuse is:

\[\frac{3}{0.5} = 6 \text{ in}\]

Additional Trigonometric Functions

In addition to the sine, cosine, and tangent functions, familiarize yourself with the secant, cosecant, and cotangent functions, which are reciprocal functions of cosine, sine, and tangent, respectively..

The secant function is defined as \(\frac{1}{\cos\theta}\). Given that the cosine function is the adjacent side divided by the hypotenuse, the secant function is the hypotenuse divided by the adjacent side.

The cosecant function is defined as \(\frac{1}{\sin\theta}\). Given that the sine function is the opposite side divided by the hypotenuse, the cosecant function is the hypotenuse divided by the opposite side.

The cotangent function is defined as \(\frac{1}{\tan\theta}\). Given that the tangent function is the opposite side divided by the adjacent side, the cotangent function is the adjacent side divided by the opposite side.

Inverse Trig Functions

The inverse of each trig function, written as \(\arcsin(x), \; \arccos(x), \text{ and } \arctan(x)\), can be used in order to “undo” the original trig function, leaving us with the angle measure. For example, \(\arcsin {\sin \theta} = \theta\). The inverse trig functions are typically used when we know the ratio between two side lengths of a right triangle, say opposite and hypotenuse, and want to find the corresponding angle. Below, we look at such an example.

Note: Sometimes, you will see the inverse trig functions written as \(\sin^{-1}(x), \cos^{-1}(x),\) and \(\tan^{-1}(x)\).

Consider a right triangle with side lengths \(3\), \(4\), and \(5\) (the unit of length doesn’t matter). Because all side lengths are known, any angle can be found by solving any trig function for \(\theta\). This is accomplished by evaluating the inverse trig function. So, suppose we want to find the angle measure of the angle opposite from the side of length \(3\). Let’s call this angle \(\theta\). Then, we know:

\[\sin \theta = \frac{o}{h} = \frac{3}{5}\]

In order to solve this problem, \(\theta\) must be isolated; to do so, we evaluate the inverse of the sine function, \(\arcsin\), of both sides:

\[\arcsin (\sin \theta) = \arcsin (\frac{3}{5})\] \[\theta = \arcsin (\frac{3}{5})\]

This can be evaluated using a scientific calculator.

The same method can be used to evaluate for an unknown side length when given an angle measurement and a side length.

Working with Circles and Trigonometry

One of the best ways to illustrate and understand the concepts of trigonometry is with the use of the unit circle. The unit circle is a circle with a radius of \(1\) that connects the coordinate plane, right triangles, trigonometric functions, degree measurement, and radian measurements. When fully described, it shows a circle divided into a collection of right triangles of various side lengths, with a vertex lying on the edge of a circle. The coordinates of each vertex along the circle represent cosine and sine values that satisfy the Pythagorean formula, \(x^2 + y^2 = 1\), or in trigonometric form, \((\cos x)^2 + (\sin y)^2 = 1\).

The vertices of these triangles also correspond with degree and radian measurements ranging from \(0\) to \(2\pi\) or \(360^{\circ}\).

This concept can be difficult to visualize, so we’ve included a series of graphics to illustrate the different aspects of the unit circle. To start, this is the basic unit circle:

The following image is the unit circle with a right triangle inscribed. Note that one of the angles starts at the circle’s center and the hypotenuse is the radius:

In this next graphic, the unitary trigonometric equation, \(\cos^{2}t + \sin^{2}t = 1\), has been added:

The following graphic shows numerous degree and radian values of points in the unit circle and their respective coordinates:

You don’t have to memorize these, but it could be helpful if you know these values for the four cardinal points: (\(0^{\circ}, 90^{\circ}, 180^{\circ},\) and \(270^{\circ}\)).