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Pythagorean Identities
Introduction: In this lesson, three trigonometric identities will be derived and applied. These involve squares of the basic trig functions and are know as the Pythagorean Identities.

The Lesson:
In a right triangle, one angle is and the side across from this angle is called the hypotenuse. The two sides which form the 90º angle are called the legs of the right triangle. We show a right triangle below. The legs are defined as either “opposite” or “adjacent” (next to) the angle A.


We shall abbreviate opposite as “opp,” adjacent as “adj,” and hypotenuse as “hyp.”

Using the Pythagorean Theorem we have
Dividing both sides of this equation by and using the relationships
sin(A) = and cos(A) = results in
Continuing, if we divide both sides of by we have
which simplifies to
.
Finally, if we divide both sides of  by we have
which simplifies to
Summary:
The three Pythagorean Identities are:
These can be rewritten by subtracting from both sides:
Identities:
It is not as easy to find angle and side measurements of triangles using the reciprocal functions as it is to use sine, cosine, and tangent. This is primarily because the calculators typically have sin, cos, and tan keys and not keys for the reciprocal functions. However, we can use these reciprocal functions to provide identities which are equations relating trig functions to each other.

An example of an identity with the variable x is
2x(3 – x) = 6x – 2x2.

This statement is true for ALL values of x.
An identity involving trig functions is
. This statement is true for any angle A.
A second example of a trig identity is
tan(A)csc(A) = sec(A).

To show that this statement is true, substitute and simplify.

tan(A)csc(A) = .
Let's Practice:
  1. In a triangle ABC, with B a right angle, suppose the sin(A) = 0.25. What is the cos(A)?
We know that .

Therefore .

This gives us cos(A) = .

In this case we took the positive square root since trig function values are positive for angles less than 90º.
  1. In a triangle XYZ with Y a right angle, suppose the tan(X) = 3. What is cot(X)? What is sec(X)?
We know that cot(X) = .

We did not need a Pythagorean Identity for this.

We also know that

Therefore .
  1. If sin(A) = 0.3 and angle A is a second quadrant angle, what is cos(A)? What is tan(A)?
We know that .

Therefore .

This gives cos(A) = .

We take the negative square root since cosine is negative for a second quadrant angle. To assist you with making these decisions on signs, refer to this lesson on reference angles.

To find the tan(A) we use tan(A) =
  1. Simplify (1 – sin(A))(1 + sin(A)).
We use the fact that to multiply (1 – sin(A))(1 + sin(A)).

The result is .

Therefore (1 – sin(A))(1 + sin(A)) = cos2(A).
  1. Show that .
We start by getting a common denominator for the fractions: .

Then we can write .

Examples
Example In a right triangle, one acute angle has a cosine of . What are the sine and cotangent of this angle?
What is your answer?
 
Example A quadrant three angle has a cosine of . What are the sine and tangent of this angle?
What is your answer?
 
Example Simplify (sec(a) – tan(a))(sec(a) + tan(a)).
What is your answer?
 
Example Show that .
What is your answer?
 



M Ransom

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