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Inverse Trig Functions
Sine Inverse Function
Sine function - An One-to-One Function
Let's take a closer look at the region of the sin function, that is one-to-one. (Right --->
If you restrict f(x) = sin x for the interval -π/2 ≤ x ≤ π/2, the function increases.
This implies that the function is one-to-one and will have an inverse.
The inverse is called the inverse sine or arcsin function.
Denoted as follows: arcsin(x) or sin-1 (x).
Remember that the second form is not the reciprocal.
Inverse Function Properties for Sin x
y = arcsin(x) can be rewritten as sin y = x.
Think of this as y is the angle and x is the answer.
Find y = arcsin(½)
Rewrite as sin(y) = ½
What angle has sine equal to ½?
The answer is π/6
Therefore, y = arcsin(½) = π/6
The fact that sin and arcsin are inverse functions can be expressed by the following equations.
sin(arcsin(x)) = x for -1 ≤ x ≤ 1
arcsin(sin(x)) = x -π/2 ≤ x ≤ π/2
Arcsin(x) domain: -1 ≤ x ≤ 1
Arcsin(s) range: -π/2 ≤ x ≤ π/2
Inverse Cosine Function
Cosine Function - An One-to-One Function
If you restrict f(x) = cos x for the interval 0 ≤ x ≤ π, the function decreases.
This implies that the function is one-to-one and will have an inverse.
The inverse is called the arccos function.
Denoted as follows: arccos(x) or cos-1(x).
Again, remember cos-1(x) is not the reciprocal.
Examples 3 and 4
Work through the following examples.
The "Try These" exercises will check your understanding of these concepts.
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