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Adding 0 to any number always results in the number itself.
Because of this, 0 is called the Additive Identity.
Any number multiplied by 1 is equal to the number itself.
Because of this, the number 1 is called the Multiplicative Identity.
For any angle θ, the following trigonometric identities hold true.
Definition | Substitute | Simplify | |
---|---|---|---|
sinθ | HypotenuseLength of opposite side to ∠θ | 1opp | opp |
cosθ | HypotenuseLength of adjacent side to ∠θ | 1adj | adj |
It can be seen that if the hypotenuse of a right triangle is 1, the sine of an acute angle is equal to the length of its opposite side. Similarly, the cosine of the angle is equal to the length of its adjacent side.
By the Pythagorean Theorem, the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. Therefore, for the above triangle, the sum of the squares of sinθ and cosθ is equal to the square of 1.
LHS/cos2θ=RHS/cos2θ
Write as a sum of fractions
aa=1
Write as a power
bmam=(ba)m
cosθsinθ=tanθ
cosθ1=secθ
Commutative Property of Addition
LHS/sin2θ=RHS/sin2θ
Write as a sum of fractions
aa=1
Write as a power
bmam=(ba)m
sinθcosθ=cotθ
sinθ1=cscθ
1+cot2θ=csc2θ
The first identity can be shown using the unit circle and the Pythagorean Theorem. Consider a point (x,y) on the unit circle in the first quadrant, corresponding to the angle θ. A right triangle can be constructed with θ.
By the Pythagorean Theorem, the sum of the squares of x and y equals 1.When a binomial is squared, the resulting expression is a perfect square trinomial.
(a+b)2=a2+2ab+b2(a−b)2=a2−2ab+b2
For simplicity, depending on the sign of the binomial, these two identities can be expressed as one.
(a±b)2=a2±2ab+b2
This rule will be first proven for (a+b)2 and then for (a−b)2.
a2=a⋅a
Distribute (a+b)
Distribute a
Distribute b
Commutative Property of Multiplication
Add terms
a2=a⋅a
Distribute (a−b)
Distribute a
Distribute -b
Commutative Property of Multiplication
Subtract terms