If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Course: Trigonometry > Unit 1

Lesson 2: Introduction to the trigonometric ratios
Math
Trigonometry
Right triangles & trigonometry
Introduction to the trigonometric ratios
© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

Trigonometric ratios in right triangles

Google Classroom
Learn how to find the sine, cosine, and tangent of angles in right triangles.
The ratios of the sides of a right triangle are called trigonometric ratios. Three common trigonometric ratios are the sine (sin), cosine (cos), and tangent (tan). These are defined for acute angle A below:
Triangle A B C with angle A C B being ninety degrees. Angle B A C is the angle of reference. Side B C is labeled opposide. Side A C is labeled adjacent. Side A B is labeled hypotenuse.
In these definitions, the terms opposite, adjacent, and hypotenuse refer to the lengths of the sides.

SOH-CAH-TOA: an easy way to remember trig ratios

The word sohcahtoa helps us remember the definitions of sine, cosine, and tangent. Here's how it works:
Acronym PartVerbal DescriptionMathematical Definition
SOHSine is Opposite over Hypotenusesin⁡(A)=OppositeHypotenuse
CAHCosine is Adjacent over Hypotenusecos⁡(A)=AdjacentHypotenuse
TOATangent is Opposite over Adjacenttan⁡(A)=OppositeAdjacent
For example, if we want to recall the definition of the sine, we reference SOH, since sine starts with the letter S. The O and the H help us to remember that sine is opposite over hypotenuse!

Example

Suppose we wanted to find sin⁡(A) in △ABC given below:
Triangle A B C with angle A C B being ninety degrees. Angle B A C is the angle of reference. Side B C is three units. Side A C is four units. Side A B is five units.
Sine is defined as the ratio of the opposite to the hypotenuse (SOH). Therefore:
Triangle A B C with angle A C B being ninety degrees. Angle B A C is the angle of reference. Side B C is three units. Side A C is four units. Side A B is five units. Sides A B and B C are highlighted.
sin⁡(A)= opposite  hypotenuse=BCAB=35
Here's another example in which Sal walks through a similar problem:
Khan Academy video wrapper
Trigonometric ratios in right trianglesSee video transcript

Practice

Triangle 1: △DEF
Triangle D E F with angle E D F being ninety degrees. Side D F is twelve units. Side E F is thirteen units. Side D F is five units.
cos⁡(F)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

sin⁡(F)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

tan⁡(F)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Triangle 2: △GHI
Triangle G H I with angle G I H being ninety degrees. Side H I is fifteen units. Side I G is eight units. Side H G is seventeen units.
cos⁡(G)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

sin⁡(G)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

tan⁡(G)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Challenge problem
In the triangle below, which of the following is equal to ac?
A right triangle with a ninety-degree angle, a twenty-degree angle, and seventy-degree angle. The side opposide of the twenty-degree angle is a units. The side opposite of seventy-degree angle is b units. The side opposite of the ninety-degree angle is c units.
Choose all answers that apply: وہ سب سلیکٹ کریں جو مناسب ہے

Want to join the conversation?

Log in
No posts yet.
Do you understand English? Click here to see more discussion happening on Khan Academy's English site.