by The memory device SOHCAHTOA helps young mathematicians remember the trigonometric functions sin (sin), cosine (cos) and tangent (tan), which they must solve for missing triangle sides and angles.
But to really understand how useful this mnemonic tool is, you first need to refresh yourself on the basics of right triangles, which are the main focus of trigonometry.
What are Design Triangles?
A right triangle, also called a right triangle, has one angle that is exactly 90 degrees. The side opposite this right angle is the longest side, called the hypotenuse.
The other two sides are referred to as the opposite side and the adjacent side, depending on their relationship to the angle being measured or referenced.
Trigonometric Functions and Ratios
Mathematicians need a way to solve for the angles of a triangle and relate them to the length of its side. To do this, they use the basic mathematical functions known as trigonometric functions, which have applications throughout science, engineering and everyday life.
Defined based on the ratios of the sides in a right triangle, the trigonometric functions are:
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Sine (sin θ): The ratio of the length of the opposite side to the hypotenuse (O/H).
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Path (cos θ): The ratio of the length of the adjacent side to the hypotenuse (A/H).
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tangent (tan θ): The ratio of the distance between the opposite side and the adjacent side (O/A).
These ratios depend on the angle θ, an acute angle (less than 90 degrees) in the triangle.
What Does SOHCAHTOA Stand For?
This is how the SOHCAHTOA mnemonic device helps mathematicians and math students remember trigonometric functions and ratios.
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SOH: Sine = Opposite / Hypotenuse
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CAH: Cosine = Adjacent / Hypotenuse
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TOA: Tangent = Opposite / Adjacent
Using the Trigonometric Ratios
Mathematicians have to calculate the unknown sides or angles in a convex triangle all the time. To do this, they apply the trigonometric functions.
For example, if you know the value of the angle θ, you can find the two sides of a right-angled triangle.
Trigonometric Function Example
Suppose you have a right triangle with:
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Angle θ = 30 degrees
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Adjacent side = a = 5
You want to find the length of the other side b.
The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side, so:
tan(θ) = opposite/adjacent
You know:
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θ = 30
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Adjacent side a = 5
So, using the tangent function looks like this:
You know from trigonometric tables or by using a calculator:
tan(30) = 1/(√3) ≈ 0.577
So:
Now, get b:
The length of the other side b about 2.885 units.
Using the Inverse Triangular Ratios
If you know the lengths of two sides of a triangle, you can find the angle θ using the inverse triangle functions:
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sin-1 (also called arcsin) for sine
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foot-1 (also called arccos) for cosine
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tan-1 (also called arctan) for tangent
Inverse Trigonometric Function Example
Suppose you have a right triangle with:
You want to find the angle θ.
The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side, so:
tan(θ) = opposite/adjacent = b/a = 3/5
To find the angle θ, you can use one of the inverse trigonometric functions:
Using a calculator, you can then determine:
θ ≈ tan-1(0.6) ≈ 30.96
In other words, the angle θ is about 30.96 degrees, which also makes it an acute angle.
Theorem of Pythagoras
Another essential concept in trigonometry is the Pythagorean theorem, coined by Pythagoras, a philosopher in Ancient Greece. He says:
where a and b are the lengths of the legs (far side and near side), and c is the diagonal.
The Pythagorean theorem is useful when you know both sides of a right triangle and you want to find a missing side.
Solution for Missing Sides and Angles
To solve the missing sides, you can rearrange the trigonometric equations. For example:
sinθ = opposite/hypotenuse ⟹ opposite = sinθ × hypotenuse
Using a calculator, you can calculate the sine, cosine or tangent of the angle θ, and then multiply it by the length of the side to get the unknown side.
Sample Calculation
Consider a right triangle where the hypotenuse is 10 units, and the angle θ is 30 degrees. To find the other side, you would use the sine function:
sin30° = opposite/10 ⟹ opposite = 10 × sin30°
From sin30° =0.5:
Triangular Identities
Trigger identities are equations involving trigonometric functions that are true for all values of the variable. They include fundamental identities such as this fundamental Pythagorean identity:
sin2(θ) + cos2(θ) =1
Other common identities include angle sum and difference identities, double angle identities, half angle identities, reciprocal identities and common function identities. These identities are useful when trying to simplify trigonometric expressions and solve equations.
Real world applications
It is easy to see trigonometry as a theoretical concept, but that is far from the case. It actually has practical applications across many different fields, including engineering, physics and astronomy, and even in everyday problem-solving situations like navigation and construction.
Engineers use trigonometric ratios to analyze forces in structures such as bridges and buildings, or to design mechanical systems in engines. With these ratios, physicists can determine the path, range, and height of objects in projectile motion, and astronomers can calculate the angles needed to precisely align their telescopes with specific celestial objects or events.
We created this article in collaboration with AI technology, then made sure it was fact-checked and edited by a HowStuffWorks editor.
Basic: How to Use the Remembering ‘SOHCAHTOA’ in Trigonometry
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