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SOHCAHTOA is a mnemonic device that is used in mathematics to remember the definitions of the three most common trigonometric functions. This lesson will explain each one and give examples, and you’ll have the opportunity to take a quiz at the end to solidify what you’ve learned.
Trigonometric Functions
Sine, cosine, and tangent are the three main functions in trigonometry. They’re all based on ratios obtained from a right triangle. Before we can discuss what ratios work for which function, we need to label a right triangle.
Opposite is the side opposite the angle in question, adjacent is the side next to the angle in question, and the hypotenuse is the longest side of a right triangle. The hypotenuse is always opposite the right angle.
The ratios that allow you to determine the sine, cosine, and tangent of a right triangle are:
The sine of an angle is equal to the side opposite the angle divided by the hypotenuse.
The cosine of an angle is equal to the side adjacent to the angle divided by the hypotenuse.
The tangent of an angle is equal to the side opposite the angle divided by the side adjacent to the angle.
SOHCAHTOA
These ratios can be difficult to remember. You might easily get confused and not remember which side goes where. SOHCAHTOA is a mnemonic device helpful for remembering what ratio goes with which function.
SOH = Sine is Opposite over Hypotenuse
CAH = Cosine is Adjacent over Hypotenuse
TOA = Tangent is Opposite over Adjacent
With these properties, you can solve almost any problem related to finding either a side length or angle measure of a right triangle. SohCahToa can ensure that you won’t get them wrong.
Examples
Now, for some example problems. Let’s find x for this triangle.
We know the side adjacent to the known angle of 60 degrees is 13 cm. We’re trying to find the length of the side opposite the known angle of 60 degrees. Thus, we need to use TOA, or tangent (tan), which uses opposite and adjacent. Therefore, our equation will be:
tan 60 = x/13
The tan of 60 is 1.73, which makes the equation:
1.73 = x/13
Solve for x to get:
x = (1.73) * (13) = 22.49
So, the length of side x is 22.49 cm.
On to another example problem. What is the sine of 35 degrees?
The sine of an angle is equal to the opposite side divided by the hypotenuse.
sin 35 = 2.8 / 4.9 = 0.57
And for one final problem, find sin, cos and tan for the angle in the right triangle shown.
sin = opposite/hypotenuse = 3 / 5 = 0.6
cos = adjacent/hypotenuse = 4 / 5 = 0.8
tan = opposite/adjacent = 3 / 4 = 0.75
Real-Life Examples
Trigonometric functions are important for many reasons. They let you calculate angles when you know sides and calculate sides when you know angles. This can be helpful in many real life situations, such as when determining the height of a large building or the distance across a lake – things that just can’t be measured easily.
 
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