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SOH-CAH-TOA

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SOH-CAH-TOA – Introduction

SOH-CAH-TOA is a mnemonic acronym used in trigonometry to remember the relationships between the sides and angles of right triangles.

Sine (sin): The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
 Cosine (cos): The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
 Tangent (tan): The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

Using SOH-CAH-TOA, you can calculate missing sides and angles in right triangles by applying the appropriate trigonometric function.

Example:
 Given an angle and the length of one side in a right triangle, you can use sine, cosine, or tangent to find the missing side length or another angle.

SOH-CAH-TOA - Basic Examples

\begin{aligned} \sin (35) & =\frac{20}{x} \\ x & =\frac{20}{\sin (35)} \\ & =34.9 \text { (1 dp) } \end{aligned}

\begin{aligned} \tan (42) & =\frac{x}{11} \\ x & =11 \times \tan (42) \\ & =9.9 \text { (1dp) } \end{aligned}

\begin{aligned} \cos x & =\frac{16}{24} \\ x & =\cos ^{-1}\left(\frac{16}{24}\right) \\ & =48.2 \text { (1dp) } \end{aligned}

\begin{aligned} \tan x & =\frac{15}{11} \\ x & =\tan ^{-1}\left(\frac{15}{11}\right) \\ & =53.7 \text { (1dp) } \end{aligned}

\begin{aligned} \tan (35) & =\frac{x}{20} \\ x & =20 \times \tan (35) \\ & =14.0(1 d p) \end{aligned}

\begin{aligned} \cos (42) & =\frac{x}{15} \\ x & =15 \times \cos (42) \\ & =11.1 \text { (1dp) } \end{aligned}

\begin{aligned} \sin x & =\frac{5}{16} \\ x & =\sin ^{-1}\left(\frac{5}{16}\right) \\ & =18.2(19 p) \end{aligned}

\begin{aligned} \tan x & =\frac{9}{5} \\ x & =\tan ^{-1}\left(\frac{9}{5}\right) \\ & =60.9 \text { (idp) } \end{aligned}

SOH-CAH-TOA - Harder Examples

\begin{aligned} \tan y & =\frac{2}{15} \\ y & =\tan ^{-1}\left(\frac{2}{15}\right) \\ & =7.6^{\circ}\left(1 d_p\right) \\ B A D & =90+7.6 \\ & =97.6^{\circ} \end{aligned}

In order to answer this question you will notice that there are two triangles both with a common length. This common length can be found first using Pythagoras and let us call the common length “y”: 

\begin{aligned} y^2+16^2 & =20^2 \\ y^2 & =20^2-16^2 \\ y^2 & =144 \\ y & =12 \end{aligned}

The length “x” can then be found as follows: 

\begin{aligned} \sin (25) & =\frac{12}{x} \\ x & =\frac{12}{\sin (25)} \\ & =28.4 \text { (1dp) } \end{aligned}
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SOH-CAH-TOA-  Final Example

 
 

First find the length AD and we will call this length “y”. Once this has been found you can then find the length CD and then the required angle as shown below. 

Remember you need to work with two triangles and so you could you be using Pythagoras or SOH-CAH-TOA or both: 

\begin{aligned} \tan (37) & =\frac{9}{y} \\ y & =\frac{9}{\tan (37)} \\ & =11.9434 \ldots \end{aligned}
\begin{aligned} C D & =22-11.9434 \\ & =10.05659 \ldots \end{aligned}\begin{aligned} \tan x & =\frac{9}{10.05 \ldots .} \\ x & =\tan ^{-1}\left(\frac{9}{10.05 \ldots}\right) \\ & =41.8 \mathrm{ldp} \end{aligned}

Conclusion:
 SOH-CAH-TOA is a useful tool in trigonometry that allows you to calculate missing sides and angles in right triangles. By understanding the relationships between sine, cosine, and tangent, you can solve various trigonometric problems.

If you, or your parents would like to find out more, please just get in touch via email at info@exam.tips or call us on 0800 689 1272. 

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