Back To Course

Calculus: Help and Review13 chapters | 148 lessons

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Login here for access

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Jennifer Beddoe*

In mathematics, trigonometric functions are functions of angles. This lesson will describe the 6 main trigonometric functions, use them to solve problems, and give some examples. The quiz at the end of the lesson will allow you to practice what you've learned.

There are **six main trigonometric functions:**

- Sine (sin)
- Cosine (cos)
- Tangent (tan)
- Secant (sec)
- Cosecant (csc)
- Cotangent (cot)

These functions are used to relate the angles of a triangle with the sides of that triangle. Trigonometric functions are important when studying triangles and modeling periodic phenomena such as waves, sound, and light.

To define these functions for the angle *theta,* begin with a right triangle. Each function relates the angle to two sides of a right triangle. First, let's define the sides of the triangle.

- The
**hypotenuse**is the side opposite the right angle. The hypotenuse is always the longest side of a right triangle. - The
**opposite**side is the side opposite to the angle we are interested in,*theta.* - The
**adjacent**side is the side having both the angles of interest (angle*theta*and the right angle).

The relationship between the trigonometric functions and the sides of the triangle are as follows:

- sine(
*theta*) = opposite / hypotenuse - cosecant(
*theta*) = hypotenuse / opposite - cosine(
*theta*) = adjacent / hypotenuse - secant(
*theta*) = hypotenuse / adjacent - tangent(
*theta*) = opposite / adjacent - cotangent(
*theta*) = adjacent / opposite

These trigonometric functions have practical applications in surveying, building, engineering, and even medicine. Here's one practical way to use these functions to solve a problem:

*The angle of elevation of an airplane is 23 degrees, and its altitude is 2500 meters. How far away is it?*

We are trying to solve this right triangle for the hypotenuse *x.* Since the side length we know is opposite the angle we know, we can use the sine function.

Sin(23) = 2500m / *x*

*x* = 6398.3 meters

You can use these ratios to solve for any side or angle of a right triangle. The information you are given will help you determine which function to use.

Example:

*Solve for b if you know that c is 2.5 km and B is 15.7 degrees.*

Since we know the measurements of the angle opposite the side we are trying to find and the hypotenuse, we can use either the sine or cosecant functions. Most often when solving these problems, the sine, cosine, and tangent functions are used because they are easier to calculate with a calculator. So, we will use the sine function for this problem.

Sin(15.7) = *b* / 2.5 km

0.271 = *b* / 2.5 km

*b* = 0.6765 km

Try this one:

*Solve triangle ABC given that A is 35 degrees and c is 15 feet.*

We don't know much about this triangle, but because it is a right triangle and we know at least two other sides or angles, we can use trigonometric functions to solve for the rest. The easiest place to start is to find the angle *B.* Since all triangles have angle measures that add up to 180 degrees, to solve for *B,* just subtract.

180 - 90 - 35 = *B*

*B* = 55 degrees

Then we can use sine and cosine to solve for sides *a* and *b.* Using angle *A,* and the hypotenuse, the equation to solve for side *a* is:

Sin(35 degrees) = *a* / 15 feet

*a* = 8.6 feet

The equation to solve for side *b* is:

Cos(35 degrees) = *b* / 15 feet

*b* = 12.3 feet

The **six main trigonometric functions** are sine, cosine, tangent, secant, cosecant, and cotangent. They are useful for finding heights and distances, and have practical applications in many fields including architecture, surveying, and engineering.

As soon as you've reviewed the lesson, apply your knowledge in order to:

- State the six trigonometric functions
- Name the sides of a triangle
- Recognize the relationships between triangular sides and trigonometric functions
- Use trigonometric functions to solve problems

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
6 in chapter 3 of the course:

Back To Course

Calculus: Help and Review13 chapters | 148 lessons

- How to Solve Visualizing Geometry Problems 10:41
- How to Calculate the Volumes of Basic Shapes 7:17
- Finding Distance with the Pythagorean Theorem 6:54
- Trigonometry: Sine and Cosine 7:26
- Trigonometry and the Pythagorean Theorem 4:14
- Trigonometric Functions: Definition & Examples 6:40
- Go to Geometry and Trigonometry in Calculus: Help and Review

- AFOQT Information Guide
- ACT Information Guide
- Computer Science 335: Mobile Forensics
- Electricity, Physics & Engineering Lesson Plans
- Teaching Economics Lesson Plans
- FTCE Middle Grades Math: Connecting Math Concepts
- Social Justice Goals in Social Work
- Developmental Abnormalities
- Overview of Human Growth & Development
- ACT Informational Resources
- AFOQT Prep Product Comparison
- ACT Prep Product Comparison
- CGAP Prep Product Comparison
- CPCE Prep Product Comparison
- CCXP Prep Product Comparison
- CNE Prep Product Comparison
- IAAP CAP Prep Product Comparison

- How to Write a Newspaper Article
- Anthem by Ayn Rand: Book Summary
- Field Hockey: Techniques, Rules & Skills
- What's the Difference Between Polytheism and Monotheism?
- The Very Hungry Bear Literacy Activities
- Totem Pole Project Ideas
- The Worst Day of My Life Ever by Julia Cook Activities
- Quiz & Worksheet - Who, What, When, Where & Why
- Quiz & Worksheet - American Ethnic Groups
- Quiz & Worksheet - Kinds of Color Wheels
- Quiz & Worksheet - Phenol Reactions
- Analytical & Non-Euclidean Geometry Flashcards
- Flashcards - Measurement & Experimental Design
- Learning Styles Guide
- 11th Grade Math Worksheets & Printables

- AP US Government and Politics: Exam Prep
- AEPA Political Science/American Government (AZ006): Practice & Study Guide
- History of the Vietnam War: Certificate Program
- Human Growth & Development Studies for Teachers: Professional Development
- Animal Behavior Study Guide
- Praxis Biology & General Science: Chemistry Review: Chemical Reactions
- Flowing Water
- Quiz & Worksheet - Gravitational Pull of the Earth
- Quiz & Worksheet - Female Reproductive Diseases & Disorders
- Quiz & Worksheet - Overview of England's Population Trends & Outlook
- Quiz & Worksheet - Regions of the World
- Quiz & Worksheet - Ancient Mali Government & Economy

- Effect of Humans on Streams
- The Odyssey Book 4: Summary & Quotes
- Accessing Results from the QTS Numeracy Skills Test
- 4th Grade Science Projects
- Boston Tea Party Lesson Plan
- Pennsylvania Homeschool Laws
- How to Prepare for the GMAT
- Scientific Method Experiments for Kids
- Shays' Rebellion Lesson Plan
- Illinois TAP Test Registration & Dates
- Georgia Science Standards
- What is the STAAR Test?

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject