SOHCAHTOA | Meaning, Formula, and Applications in Trigonometry

Last Updated : 23 Jul, 2025

SOHCAHTOA (pronounced as "soh-kah-toe-ah") is a simple way to remember the trigonometry rules for a right-angled triangle. The trigonometry ratios are: sine(sin), cosine(cos), and tangent (tan).

The calculation simply involves one side of a right-angle triangle divided by another side. We need to know which side, and that is where SOHCAHTOA helps.

Below is the visual representation of the SOHCAHTOA triangle:

SOHCAHTOA
SOHCAHTOA Triangles

Table of Content

SOHCAHTOA Formula

  • SOH: Sine(sin) = Opposite / Hypotenuse
  • CAH: Cosine(cos) = Adjacent / Hypotenuse
  • TOA: Tangent(tan) = Opposite / Adjacent

This helps you easily remember which sides of the triangle correspond to each trigonometric function. Here’s a breakdown:

  • Sine uses the opposite side and the hypotenuse.
  • Cosine uses the adjacent side and the hypotenuse.
  • Tangent uses the opposite side and the adjacent side.

How to label the sides of a right-angle triangle?

Right-angled-Triangle
Right Angled Triangle
  • The hypotenuse (AC) is the longest side of the triangle. It is the opposite side of a right angle.
  • The opposite side (AB) is the side opposite side to the angle.
  • The adjacent side (BC) is next to the angle.

When to use SOHCAHTOA?

To use SOHCAHTOA, we need to know certain measures of a right-angle triangle:

  • We can use SOHCAHTOA if we know two sides of the triangle.
  • We can use SOHCAHTOA if we know one side and one angle of a right angle.

SOHCAHTOA Memory Summary:

  • Students Our Homework Can Help To Overcome Algebra.
  • Silly Owls Hide Cake And Honey Till October Arrives.

Find Unknown angles using SOHCAHTOA.

Calculate the side of the right angle labeled as x.

Untitled-design
Triangle

Step 1: Label the sides of the right triangle with respect to one of the acute angles.

  • The side marked as 9 cm is next to angle, so it is adjacent side.
  • The side marked as 10 cm is hypotenuse because it is across from the right angle.

Step 2: Finding the angle θ using trigonometry:

To find angle θ, we will use the cosine function.

  • According to SOHCAHTOA, the cosine function (C) is the ratio of the Adjacent side (A) to the Hypotenuse(H):
    cos θ = Adjacent (A) ​/ Hypotenuse (H)
  • As we are finding angle ,use the inverse cosine function.
    θ = cos -1 = ( A / H)
  • Substitute the two known side length into the formula.
    θ = cos -1 ( 9/ 10 )
  • Using a calculator to find the inverse cosine:
    θ = cos-1 ( 9/ 10 ) = 25.84

So, the missing angle θ is approximately 25.84°.

Finding the missing side

Calculate the side of the right angle labeled as x.

Triangle
Triangle

Step 1: Label the sides of the right triangle with respect to one of the acute angles.

  • The hypotenuse (H) is given as 8 cm.
  • The opposite side (O) is the side that we need to find, labeled as x.
  • The adjacent side is not relevant to this calculation.

Step 2: Determine the trigonometric ratio to use and write the formula with the correct subject.

  • Since we are dealing with the opposite side and the hypotenuse, we will use the sine function from SOHCAHTOA.
  • The sine function is sin θ = Opposite (O) / Hypotenuse (H)
  • To find the opposite side( O ), rearrange the formula by covering up the opposite side:
    O = H × sin θ

Step 3: Substitute known values into the formula.

  • The hypotenuse (H) is 8 cm, and the angle (θ) is 40°.
  • Now substitute these values into the fromula:
    O = 8 × sin (40°)
  • Using a calculator to find sin⁡(40∘):
    sin⁡(40∘) ≈ 0.6428
  • Now calculate O:
    O = 8 × 0.6428 = 5.14 cm

So, the missing opposite side x is approximately 5.14 cm.

SOHCAHTOA Inverse

The inverse trigonometric functions are used to find angles when you know the ratios of the sides of a right triangle. Here's how to apply the inverse functions based on SOHCAHTOA:

  • Inverse sine (sin⁡−1: θ = sin⁡−1 ( \dfrac{\text{Opposite}}{\text{Hypotenuse}})
  • Inverse cosine (cos⁡−1: θ = cos⁡−1 ( \dfrac{\text{Adjacent}}{\text{Hypotenuse}})
  • Inverse tangent (tan⁡−1: θ = tan⁡−1 ( \dfrac{\text{Opposite}}{\text{Adjacent}})

Mistakes to Avoid

  • Use the incorrect mode of calculator- When calculating side or angle measurement using one of the trigonometric ratios, be sure that the calculator is in degree mode, not in radian mode.
  • Using SOHCAHTOA for non-right angle - if a triangle does not have a 90-degree angle, try to use the cosine and sine rule rather than SOHCAHTOA.
  • Forgetting to use the inverse trigonometric function for angles - if solving for angles, always use the inverse trigonometric function instead of the regular trigonometric ratio.

Solved Examples on SOHCAHTOA

Question 1: Given a right-angled triangle, with base equals 6cm, perpendicular equals 8cm and hypotenuse equals 10cm. Find the value of sinθ, cosθ, and tanθ.

image
Triangle

Solution:

Given,
Opposite = 6 cm
Adjacent = 8 cm
Hypotenuse = 10 cm

sinθ = Opposite / Hypotenuse
sinθ = 6 / 10 = 0.6
cosθ = Adjacent/ Hypotenuse
cosθ = 8 / 10 = 0.8
tanθ = Opposite / Adjacent
tanθ = 6 / 8 = 0.75

The values are 0.6 , 0.8 and 0.75.

Question 2: Use the triangle below to find the values of sinθ, cosθ, tanθ, secθ, cotθ, and cosecθ.

Solution:

SOHCAHTOA-2
Triangle

First we need to find length of hypotenuse through pythagorean theorem,

(Hypotenuse) 2 = (Perpendicular) 2 + (Base) 2
C 2 = ( 15 ) 2+ ( 8 ) 2
C 2 = 225 + 64
C 2 = 289
C 2 = √ 17
C = 17

sin θ = Opposite / Hypotenuse
sin θ = 15 / 17
cos θ = Adjacent/ Hypotenuse
cos θ = 8 / 17
tan θ = Opposite / Adjacent
tan θ = 15 / 8
sec θ = Hypotenuse /Adjacent
sec θ = 17 /8
cosec θ = Hypotenuse / Opposite
cosec θ = 17 /15
cot θ = Adjacent / Opposite
cot θ = 8 / 15

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