Trigonometric Functions
Definition: Let be a right angle triangle such that B = 90o as shown below and we use lowercase letters to indicate the length of the side corresponding to its associated angle in the triangle.
Then we define the following trigonometric functions on A:
sinA = , cosA = , tanA= cscA = , secA= , cotA= .
They are known as “sine” ( sin ), “cosine” (cos), “tangent” ( tan ), “cosecant” ( csc ), “secant”( sec ), and “cotangent” ( cot ).
Please notice that we define these functions along with a right angle triangle. It limits the argument of these functions to the range between 0o to 90o .
There will be some other ways to define those functions to extend the concept such that broader range of arguments for these trigonometric functions can be allowed and well defined.
Properties: B = 90o. Thus, we have a2 + b2 = c2 . By that, we have the following equalities:
sin2A + cos2A = 1 1 + tan2A = sec2A 1 + cot2A = csc2A And by definition, we can easily get sinA cscA = 1 tanA cotA = 1 cosA secA = 1
Properties: Recall the triangle we use for definition: A + C = 90o . Similarly, we have those trigonometric functions for C. For example,
sinC = , cosC = , tanC =
Thus, we get sin( 900 – A ) = cosA, cos(900 – A ) = sinA, tan(900 – A ) = cotA The rest of relationship can be deducted in similar ways.
Properties: Every trigonometric function can be represented as the functions of sine and cosine.
tanA = . And any other trigonometric functions can be represented by sine, cosine, and tangent. So, sine and cosine can be used to represent all the other trigonometric functions.
Theorem: ( Area of Triangle ) = bc sinA Proof: It is self-clear by using the following diagram:
Remember that our current definition for trigonometric functions is limited to the angles that are less than 90o. Actually, the theorem still holds for Obtuse Triangle.
Once the relationship between sinA and sin( 180o-A ) is given, it is easy to come back to check this theorem again.
Theorem ( Sine Law )
Proof: = bc sinA = ba sinC = ca sinB
Let each be divided by abc:
= =
Thus,
Still the theorem holds for Obtuse Triangle. Once the relationship between sinA and sin( 180o-A ) is given, it is easy to come back to check this theorem again.
Another view of this equality: Consider the following diagram. Let O be the center of the circle and A, B, C are on the circle. Let r be the radius of the circle.
Please notice that BCD = 90o , and BDC = BAC Then, a = = 2r sinD
= 2r The same approach can be applied to the other angles.
Theorem ( Cosine Law ) ( Proof for Acute Triangle )
cosA = ( b2 + c2 – a2 )
Proof:
1. 2 = 2 - 2 = 2- 2
2. = b cosA 2 = b2 – ( b cosA )2 , = c – b cosA
3. Substitute the result from 2 into 1: b2 – ( b cosA )2 = a2 – (c – b cosA)2 cosA = ( b2 + c2 – a2 )
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