SOHCAHTOA | Meaning, Formula, and Applications in Trigonometry (original) (raw)

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 Triangles

Table of Content

• SOHCAHTOA Formula
• How to label the sides of a right-angle triangle?
  • When to use SOHCAHTOA?
  • Find Unknown angles using SOHCAHTOA
• Finding the missing side
  • SOHCAHTOA Inverse
• Solved Examples on SOHCAHTOA

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

• 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.

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

**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.

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:

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

**Related article

• Trigonometric ratio
• What are Sine, Cosine, and Tangent?