Sohcahtoa - Explained with 3 clever Examples.

Sohcahtoa – Explained with 3 clever Examples.

Hey, are you really tired of remembering the Trigonometric functions given on your book? The trig ratios were also hard for me to remember when I had first started my Trignometry chapter in my class.

I asked my teacher to give me any simple trick to remember these identities, and with just a smile, my teacher told me a single word, sohcahtoa.

Sohc…what? = soh cah toa (say ‘so-ka-toe-uh‘)

Well, after thinking for two hours on my own, I actually understood the meaning of the word SOH CAH TOA. It’s not rocket science to look at. It’s just a way to remember how the trigonometric functions (i.e. sin, cos, and tan) of a right triangle can be computed.

So in today’s article let us talk more about this, with suitable sohcahtoa questions –

What is SohCahToa?

Sohcahtoa is simply anything that is used to remember the definition of Trigonometric Ratios (i.e. sine, cosine and tangent) easily.

The meaning for the word can also be understood by breaking it into parts = SOH CAH TOA.

SOH CAH TOA

SOH denotes the identity for sine, where, Sin = Opposite / Hypotenuse

Like wise, CAH denotes, Cos = Adjacent / Hypotenuse

And TOA denotes, Tan = Opposite / Adjacent

What does sohcahtoa stand for?

Soh-cah-toa stands for the trick to remember the Trigonometric Functions where,

Sin = Opposite / Hypotenuse

Cos = Adjacent / Hypotenuse

Tan = Opposite / Adjacent

How to use sohcahtoa to find angles?

As you have seen in trig identities, “Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse and Tangent is Opposite over Adjacent”. So these identities might be confusing for many learners.

Thus, Soh-cah-toa is a simple way to remember these trig ratios. The use of SOH CAH TOA is also been justified by its name when we break down it.

  • Sin = Opposite / Hypotenuse
  • Cos = Adjacent / Hypotenuse
  • Tan = Opposite / Adjacent

Note: Opposite is the side opposite to angle θ, Adjacent is the side which is just next to angle θ and Hypotenuse is the longest side, opposite to the right angle.

Opposite, Adjacent and Hypotenuse of right triangle (Sohcahtoa)

So if the question is like,

Q. Find the values of sinθ, cosθ, and tanθ from the following right triangle given.

How to use sohcahtoa to find angles?

Solution:

Sin θ = Opposite / Hypotenuse

Therefore, Sin θ = 3/5 (Ans.)

Likewise, Cos θ = Adjacent / Hypotenuse

∴ Cos θ = 4/5 (Ans.)

and, Tan θ = Opposite / Adjacent

∴ Tan θ = 3/4 (Ans.)

How to use sohcahtoa to find a side?

To find a side from SOH, CAH and TOA. One length and one angle (apart from the right angle) must be known to us.

For example,

Q. Find the side xy from the given figure.

How to use sohcahtoa to find a side?

Solution:

So above, we have a figure of Δxyz in which one length and one angle is given. So to find the side xy, concede the following steps.

Step 1:

Look gently that which sides are given, out of Opposite, Adjacent and Hypotenuse in the question. Here, the hypotenuse is given by 50m and also ∠yxz is given, 36°.

Thus, we have to find the side, xy.

right?

So which identity should we use from SOH, CAH and TOA?
Ans. SOH

Step 2:

Use SOHCAHTOA to find which trig function to use in the question. Here, we’ll use SOH (i.e, sin = opp. / hyp.), as we know the value of Hypotenuse and we’re asked to find the value for the Opposite side (from θ).

Step 3:

Put all the known values in the trig ratio for Sine i.e, sin = opposite / hypotenuse, and by using algebra, solve the equation.

∴ Sin θ = opp. / hyp.

∴ Sin (36°) = xy / 50

So we have the value of Sin (36°) = 0.5878 (round off)

∴ 0.5878 = xy / 50

∴ xy = 0.5878 × 50 = 29.39

The length of the side xy is 29.39‬m. (Ans)

How do you do SOH CAH TOA problems?

SOH CAH TOA is an easy way to remember/find trigonometric identities. To solve SOH CAH TOA problems, just put the value of sides of the triangle, in SOH, CAH or TOA.

Following are some Solved sohcahtoa questions –

Sohcahtoa Questions: Solving Right Triangles with Sohcahtoa –

SOHCAHTOA QUESTION 1:

1. A rabbit sits far from a building, making an angle of 72° from its top. If the building is 60m tall. How far the rabbit is sitting, when calculated from the base of the building?
Sohcahtoa Questions

Solution:

Given,

Height of the building = 60 m and,

the angle between the adjacent side and the hypotenuse is 72°

∴ we need to find the opposite side (from the angle 72°)

So we would use TOA i.e,

∴ Tanθ = Opp. side / Adj. side

∴ Tan(72) = Opposite side / 60

∴ Tan(72) . 60 = Opposite Side

[Value of Tan(72) = 3.077]

∴ Opposite Side = 3.077 . 60 = 184.62‬ m

Therefore, the rabbit is sitting at 184.62 ‬meters distance, when calculated from the base of the building.

SOHCAHTOA QUESTION 2:

2. Find the value for X, from the figure.
solving Sohcahtoa Questions:

Solution:

Given,

Adjacent Side = 14 m and, angle = 67

To Find X (opposite side),

∴ tanθ = opp / adj

∴ tan(67) = x / 14

∴ 14 × tan(67) =x

value of tan(67) = 2.35585

∴ x = 32.98 m (Ans)

SOHCAHTOA QUESTION 3:

3. Find the value of X from the following triangle.
Sohcahtoa Questions for right triangle

Solution:

Given,

Adjacent Side = 48 ft. and, Opposite Side = 14 ft.

To Find X (Hypotenues)

Note: Here we can do this question by two methods. First by using soh, cah and toa. And another by Pythagorean Theorem.

Solving through SOHCAHTOA.

Use, Sinθ = Opp. / Hyp. (SOA)

∴ Sin(16) = 14 / Hyp

∴ Hypotenues = 14 / Sin(16)

∴ Hypotenues = 14 / 0.2756

∴ Hypotenues = 50.79 ft. (Ans.)

Solving through Pythagorean Theorem

Given,

Perpendicular (Adjacent side) = 48 ft.

Base (Opposite Side) = 14 ft.

To Find, Hypotenues –

∴ (Hypotenues)2 = (Perpendicular)2 + (Base)2

∴ Hypotenues = (Perpendicular)2 + (Base)2

∴ Hypotenues = (48)2 + (14)2

∴ Hypotenues = <SOLVE AND COMMENT THE ANSWER>

Sohcahtoa explained (Video Explanation) –

The following video from Homemade Mathematics shows the detailed explanation of Soh Cah Toa with some basic sohcahtoa examples and sohcahtoa questions.

Just click below to watch it.

Some of the basic questions asked on Google about SOH CAH TOA are given below in the form of FAQs.

FAQs

What does sohcahtoa mean?

SOH CAH TOA means the shortcut trick to remember trigonometric identities (i.e sine, cosine and tangent), where
Sin = Opposite / Hypotenuse
Cos = Adjacent / Hypotenuse
Tan = Opposite / Adjacent

How do you know when to use SOH CAH and Toa?

If in the question, the values of the hypotenuse and the opposite side (from θ) are provided, use soh i.e, Sin = Opposite / Hypotenuse. If you are working with the Adjacent side and the Hypotenuse, use CAH i.e, Cos = Adjacent / Hypotenuse. Likewise, if you have the opposite side along with the adjacent side, use TOA i.e, Tan = Opposite / Adjacent.

Can you use SOH CAH TOA for any triangle?

No, Soh-Cah-Toa is only applicable for the right triangle (triangle with 90°). As trigonometry is basically made for right-angled triangles, thus the use of soh, cah, and toa is also used for only right triangles.

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2 thoughts on “Sohcahtoa – Explained with 3 clever Examples.”

  1. Sohcahtoa was something new I have heard. This article has really helped me understand its concept and use in Trigonometry.

    Reply

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