MCR3U1
U5L8
TRIGONOMETRIC IDENTITIES
(Part 1)
PART A ~ INTRODUCTION
A trigonometric identity is an equation involving trigonometric ratios that is true for
all values of the variable ( or x).
Reciprocal Identities:
𝑐𝑠𝑐𝜃 =
𝑠𝑒𝑐𝜃 =
𝑐𝑜𝑡𝜃 =
1
𝑠𝑖𝑛𝜃
1
𝑐𝑜𝑠𝜃
Quotient Identities:
𝑡𝑎𝑛𝜃 =
𝑐𝑜𝑡𝜃 =
𝑠𝑖𝑛𝜃
𝑐𝑜𝑠𝜃
𝑐𝑜𝑠𝜃
𝑠𝑖𝑛𝜃
1
Pythagorean Identities:
𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1
1 + 𝑡𝑎𝑛2 𝜃 = 𝑠𝑒𝑐 2 𝜃
1 + 𝑐𝑜𝑡 2 𝜃 = 𝑐𝑠𝑐 2 𝜃
𝑡𝑎𝑛𝜃
NOTE: An identity can be rearranged (any) or squared (reciprocal or quotient).
PART B ~ IDENTITIES DERIVED FROM RELATIONSHIPS
Derive each of the following identities:
𝑡𝑎𝑛𝜃 =
𝑠𝑖𝑛𝜃
𝑐𝑜𝑠𝜃
𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1
r
y

x
1 + 𝑡𝑎𝑛2 𝜃 = 𝑠𝑒𝑐 2 𝜃
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PART C ~ STRATEGY TO PROVE IDENTITIES
To prove that a given trigonometric equation is an identity, both sides of the equation
need to be shown to be equivalent. This can be done by:
i)
ii)
simplifying the more complicated side until it is identical to the other side;
manipulating both sides to get the same expression.
Strategies:
1.
2.
3.
4.
5.
6.
Start with the more complicated side.
Rewrite all tangent expressions in terms of sine and cosine.
Where denominators are involved, determine a common denominator.
Use any form of the Pythagorean identity only if it simplifies the expression.
Factor if needed (F – it!)
Collect like terms.
SPECIAL NOTE:
Proving an identity is not the same as solving an equation. An identity is true for all
values of the variable whereas an equation has specific solution(s).
Solving an equation requires isolating the variable on one side of the equation whereas
proving an identity involves simplifying either side separately until they are
equivalent.
Ex.
y
Solve the equation:
2𝑠𝑖𝑛𝜃 + 1 = 0
2𝑠𝑖𝑛𝜃 = −1
𝑠𝑖𝑛𝜃 = −
√3
1
2
1
∴ 𝜃 = 2100 , 3300
PART D ~ EXAMPLES
Prove each of the following identities:
a)
𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥 𝑡𝑎𝑛𝑥 = 1 − 𝑐𝑜𝑠 2 𝑥
b)
𝑡𝑎𝑛2 𝑥 =
1−𝑐𝑜𝑠2 𝑥
1−𝑠𝑖𝑛2 𝑥
x
2
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c)
e)
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𝑠𝑖𝑛𝑥 + 𝑡𝑎𝑛𝑥 = 𝑡𝑎𝑛𝑥(1 + 𝑐𝑜𝑠𝑥)
𝑠𝑖𝑛2 𝑥
1−𝑐𝑜𝑠𝑥
d)
1
1+𝑠𝑖𝑛𝑥
+
1
1−𝑠𝑖𝑛𝑥
=
2
𝑐𝑜𝑠2 𝑥
= 1 + 𝑐𝑜𝑠𝑥
HOMEWORK: p.310–311 #5, 8