Understanding Trigonometric Identities
Have you ever noticed that some mathematical equations are always true for any value of their variables? For example, is always true for any values of and . Equations like this are called identities.
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Have you ever noticed that some mathematical equations are always true for any value of their variables? For example, is always true for any values of and . Equations like this are called identities.
In trigonometry, we also have equations that are always true for any angle value. These are called trigonometric identities. These identities are very useful for simplifying trigonometric expressions and solving equations.
Let's start with the most fundamental identity. Consider a unit circle with point that forms angle with the positive -axis.
On the unit circle:
Using the Pythagorean theorem for point :
Substituting the values of and :
Or can be written as:
This is the Pythagorean identity, the most fundamental identity in trigonometry.
Other Forms of Pythagorean Identity:
From the basic identity above, we can derive two other forms:
Second form: Divide both sides by (for )
Third form: Divide both sides by (for )
Each trigonometric function has its reciprocal. This relationship forms reciprocal identities:
Or in the opposite form:
Quotient identities relate tangent and cotangent to sine and cosine:
Both identities can be proven directly from the definition of trigonometric functions on the unit circle.
When angles are negative, trigonometric functions have special properties:
Even function (symmetry about -axis):
Odd functions (symmetry about origin):
Let's see how trigonometric identities are used to prove other equations.
Simplify
Solution:
Prove that
Solution:
We start from the left side:
It is proven that the left side equals the right side.
Trigonometric identities are very useful for determining the values of all trigonometric functions when one of them is known.
If and (quadrant II), determine the values of other trigonometric functions.
Solution:
Use the Pythagorean identity to find :
Since is in quadrant II, then . Therefore,
Next, calculate the other trigonometric functions:
Simplify the expression
Prove the identity
If and , determine the values of all trigonometric functions.
Simplify
If and , determine the values of and .
Let's simplify step by step:
To prove the identity, we will transform the left side:
Given in quadrant IV.
Finding :
Use difference of squares factoring:
Given and .
Since and , then (quadrant III).
Other trigonometric functions:
Use the identity :
For :