Question

How do I know when to use Soh Cah Toa or Pythagoras?

Answer 1

Hint: First know the meanings of these identities and then consider the cases where you can use them during any of the trigonometric operations. We use the Pythagoras when we have a right-angled triangle, and two sides are known. Whereas Soh Cah Toa is just a memorizing technique to remember the definitions of the trigonometric functions easily.

Complete step-by-step solution:
Firstly, let us discuss Soh Cah Toa.
It is just memorizing the definitions of the trigonometric functions.
The first word which is Soh depicts,
S for Sine,
O for the opposite,
H for the hypotenuse.
And the definition of Sine is the ratio of opposite upon hypotenuse.
In a similar way,
The second word which is Cah depicts,
C for Cos,
A for the adjacent,
H for the hypotenuse.
And the definition of Cos is the ratio of adjacent upon hypotenuse.
In a similar way,
The third word which is Toa depicts,
T for Sine,
O for the opposite,
A for the adjacent.
And the definition of Tan is the ratio of opposite upon adjacent.
The Pythagoras theorem states that,
The third side of the triangle can be found, given a right-angled triangle and two other sides.
Let us consider the right triangle whose sides are of length, a, b, and c.

So according to Pythagoras,
${{a}^{2}}+{{b}^{2}}={{c}^{2}}$
Hence, we can find any side from the above formula, given two other sides of the right-angled triangle.
We use this to find the sides which in turn can be put into the trigonometric formula.

Note: If one unknown side and two known sides of the right-angled triangle are given and asked to find the value of the trigonometric function which contains the unknown length, we first find the length of that side using Pythagoras theorem and then find the value by putting the values of that particular trigonometric function in the Soh Cah Toa formula.