Online Trig Cheat Sheet

Trigonometry Sample Exam Questions

Click for YouTube Video Solution ==>











Sample Exam #1 - Right Triangle Trigonometry
Sample Exam #2 - Circular Motion and Graphing
Sample Exam #3 - Multiple Angles and Solving
Sample Exam #4 - General Triangles and Vectors
Sample Final Exam - Cumulative Exam including Polar Graphing and Complex Numbers

Using Radians

Because radian measure is the ratio of two lengths, it is a unitless measure. For example, suppose the radius were 2 inches and the distance along the arc were also 2 inches. When we calculate the radian measure of the angle, the “inches” cancel, and we have a result without units.
\( \require{cancel}\)
\[\theta \, \text{radians} = \frac{s}{r} =  \frac{2 \cancel{\text{in.}}}{2 \cancel{\text{in.}}} = 1 \]
Therefore, it is not necessary to write the label “radians” after a radian measure, and if we see an angle that is not labeled with “degrees” or the degree symbol, we can assume that it is a radian measure.

Considering the most basic case, the unit circle (a circle with radius 1), we know that 1 rotation equals 360 degrees, 360°. We can also track one rotation around a circle by finding the circumference, C=2πr, and for the unit circle C=2π. These two different ways to rotate around a circle give us a way to convert from degrees to radians.
\[ \begin{align}
1\, \text{rotation} &= 360^\circ = 2\pi \,\text{radians} \\
 \frac{1}{2}\, \text{rotation} &= 180^\circ = \pi \,\text{radians} \\
 \frac{1}{4}\, \text{rotation} &= 90^\circ = \frac{\pi}{2} \,\text{radians}
\end{align}\]
Identifying Special Angles Measured in Radians
In addition to knowing the measurements in degrees and radians of a quarter revolution, a half revolution, and a full revolution, there are other frequently encountered angles in one revolution of a circle with which we should be familiar. It is common to encounter multiples of 30, 45, 60, and 90 degrees. Memorizing these angles will be very useful as we study the properties associated with angles. Here, we can list the corresponding radian values for the common measures.

YouTube Video: Radian Measure

Attribution: OpenStax CAT, Algebra and Trigonometry. OpenStax CNX. Aug 11, 2015 http://cnx.org/contents/13ac107a-f15f-49d2-97e8-60ab2e3b519c@5.189.

Right Triangle Trigonometry Definitions

Given triangle ABC,

There are six possible side ratios named: Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent.

The Pythagorean theorem applies,

Special 30-60-90 triangle:

Special 45-45-90 triangle:

Special Trig Function Values:

Pythagorean triples (primitives) with c ≤ 100:

Note: (6, 8, 10) is not a primitive because it is a multiple of (3, 4, 5).

Trigonometric functions of special angles in classic table form:

Radian Measure for Angles

Objective: Define radian measure for angles.

We define one radian (rad) as the central angle of a circle that subtends an arc equal to the length of the radius of the circle.

As you can see, one radian is quite a bit larger than one degree. In fact, one radian is approximately 57.3°.  An angle measuring two radians subtends an arc of length two times the radius of the circle, three radians subtends an arc of length 3 times the radius, and so on.  In general, the arc length can be expressed as some factor times the radius follows.

It is this factor θ, the ratio of the arc length s subtended by a central angle divided by the radius r of the circle, which defines radian measure of an angle.

For example, consider the circle of radius 2 centimeters in which central angle θ subtends an arc of length 6 centimeters as illustrated below.

The radian measure of the angle is,

Notice that the definition results in a unitless real number.  When units are preferred, we will use rad as an abbreviation for radians, however, this is not required. It is typical to consider values for angles without expressed units to be radian measure.

Certainly, angles measuring 3° and 3 radians are two very different angles.
---

Trigonometry Final Exam Questions

A sampling of Trigonometry exam questions with solutions.

1. Prove the following identity.

 2. Use a calculator to find the angle to the nearest tenth of a degree.

 

3. Write answer in terms of u.

4. Use exact values.

5. Find the exact and approximate area.

 

6. Find the missing parts.

 

7. Solve.

8. solve and round answers to the nearest tenth of a degree.

9. Solve using exact answers in radians.

First find the critical numbers (the zeros)

Now construct a sign chart using these critical values.

10. Solve.