SOH CAH TOA. Finding angles in a right angled triangle using trigonometry.

Finding An Angle Using Trigonometry Video

Provided that you know the values of two side lengths in a right angled triangle then you can use these two side lengths to find the size of an unknown angle in the triangle. First you will have to label up the right angled triangle:

Hypotenuse (the longest side length and the side opposite the right angle)

Opposite (the side opposite the angle you are finding)

Adjacent (the side next to the angle you are finding)

One this is done select one of three trigonometric formulas you need to work out the angle. Use the trig formula that uses the two side lengths already given in the question.

Sin Ѳ = 0/H

Cos Ѳ = A/H

Tan Ѳ = O/A

(Ѳ is the angle, H is the hypotenuse (longest side), O is the side opposite the angle, and A is the adjacent side). Try to remember the word SOH CAH TOA to help you remember these formulas.

All you need to do next is sub in your side lengths into your chosen trig formula which will make a fraction on the right hand side. The final step is to convert this fraction to an angle by sin inversing, cos inversing or tan inversing your fraction.

Example 1

Work out the size of angle x in this right angled triangle.

13 cm is the longest side of the triangle so label this H.

5 cm is the side opposite the angle so label this O

Therefore, since O and H are the two sides being used, the formula you need to use is Sin Ѳ = 0/H. So substitute the values H = 13 and O = 5 into this formula:

Sin Ѳ = 0/H

Sin x = 5/13

All you need to do now is sin inverse the 5/13 to convert this fraction into an angle:

x = sin⁻¹ (5/13) = 22.6⁰ to 1 decimal place.

Example 2

Work out the size of angle y in this right angled triangle.

11 cm is the side opposite the angle so label this O

2 cm is the side next to the angle and hypotenuse so label this A.

Therefore, since O and A are the two sides being used, the formula you need to use is Tan Ѳ = 0/A. So substitute the values O = 11 and A = 2 into this formula:

Tan Ѳ = 0/A

Tan y = 11/2

All you need to do now is sin inverse the 5/13 to convert this fraction into an angle:

y = tan⁻¹ (11/2) = 79.7⁰ to 1 decimal place.

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3 comments

Jake Bloor 4 years ago

Very Good - helped hugely thank you :) :)


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flagostomos 5 years ago from Washington, United States

I remember my old geometry teacher had a poster he had made himself that was the "Soh Cah Toa" tribe. Lol, gotta love Mr. Opp... anyway useful hub for sure.


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A.A. Zavala 5 years ago from Texas

I could have used these helful tips in physics class. Will you be writing about any statistical formulas? How about ANOVA's? Thank you for sharing.

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