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Trigonometric ratios in right triangles

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:
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:
Sine is defined as the ratio of the opposite to the hypotenuse (SOH). Therefore:
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
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
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?
Choose all answers that apply: وہ سب سلیکٹ کریں جو مناسب ہے