Inverse Functions and Angle Formulas

First, find x in terms of y, then switch x and y.

Because for y = 4, there are two values of x (2 and -2) that satisfy the equation, leading to ambiguity.

The Horizontal Line Test checks if a horizontal line intersects a graph at more than one point, indicating the function has no inverse.

<p>A function has an inverse if it is either strictly increasing or strictly decreasing.</p> Signup and view all the answers

<p>You can determine if f has an inverse by checking if the equation f(x1) = f(x2) implies x1 = x2.</p> Signup and view all the answers

<p>A function must have unique outputs for every input to possess an inverse function.</p> Signup and view all the answers

<p>First, express x in terms of y as x = (y + 2)^(1/3), then switch to get y = (x + 2)^(1/3).</p> Signup and view all the answers

<p>If two different x-values yield the same output, the function does not have an inverse.</p> Signup and view all the answers

<p>cos(𝛼 − 𝛽) = cos 𝛼 cos 𝛽 + sin 𝛼 sin 𝛽.</p> Signup and view all the answers

<p>It expands to 2 - 2 cos(𝛼 − 𝛽), which simplifies using the Pythagorean identity.</p> Signup and view all the answers

<p>cos(𝛼 + 𝛽) = cos 𝛼 cos 𝛽 − sin 𝛼 sin 𝛽.</p> Signup and view all the answers

<p>Combining cos² and sin² terms with Pythagorean identities results in a simplified expression.</p> Signup and view all the answers

<p>The negative sign indicates the sine function of a negative angle, defining its value as a negative.</p> Signup and view all the answers

<p>They equate to 2 − 2(cos 𝛼 cos 𝛽 + sin 𝛼 sin 𝛽).</p> Signup and view all the answers

<p>The basic identity applied is cos²θ + sin²θ = 1.</p> Signup and view all the answers

<p>It is treated as −sin(𝛽) due to the properties of the sine function.</p> Signup and view all the answers

<p>sin^(-1)x can refer to the inverse sine function or to csc(x), while arcsin(x) specifically denotes the inverse sine function with restricted domains.</p> Signup and view all the answers

<p>They are not pure inverses as they only apply to outputs within restricted domains, which ensures unique values.</p> Signup and view all the answers

<p>arcsin(sin(7)) = 7 - 2π because 7 is outside the acceptable range of the arcsin function.</p> Signup and view all the answers

<p>sin(arcsin(7)) = DNE (Does Not Exist) because 7 exceeds the range of the sine function (-1 to 1).</p> Signup and view all the answers

<p>The range of arcsin is restricted to $[-π/2, π/2]$.</p> Signup and view all the answers

<p>The equation holds true when b = 7 - 2π, since this value lies within the arcsin range.</p> Signup and view all the answers

<p>An angle is valid for an inverse trigonometric function if it falls within the function's specific range.</p> Signup and view all the answers

<p>csc(x) is the reciprocal of sin(x), defined as 1/sin(x).</p> Signup and view all the answers

<p>It has a removable discontinuity at $x = 0$.</p> Signup and view all the answers

<p>If $f$ is a continuous function on the interval $[a, b]$, then it takes every value between $f(a)$ and $f(b)$.</p> Signup and view all the answers

<p>Because a continuous function cannot be broken, meaning you cannot lift your pencil while drawing it.</p> Signup and view all the answers

<p>At $f(0) = -1$ and $f(\frac{\pi}{2}) = \frac{\pi}{2}$, by the IVT, since $-1 &lt; 0 &lt; \frac{\pi}{2}$, there is a $c$ where $f(c) = 0$.</p> Signup and view all the answers

<p>The IVT ensures that for every positive $eta$ between $0$ and $\zeta$, there is a corresponding $c$ that satisfies $c^n = \beta$.</p> Signup and view all the answers

<p>$f(x)$ is continuous since both $x$ and $\cos(x)$ are continuous functions, enabling the application of the IVT.</p> Signup and view all the answers

<p>The limits as $x$ approaches 0 highlight the removable discontinuity and indicate the limit is 1.</p> Signup and view all the answers

<p>It allows us to show that functions attain certain values over an interval, which is fundamental in analysis.</p> Signup and view all the answers

<p>The general formula is given by $y - y_0 = k(x - x_0)$.</p> Signup and view all the answers

<p>It begins by selecting another point, x, and calculating the slope of the line through (a, f(a)) and (x, f(x)).</p> Signup and view all the answers

<p>The purpose is to obtain the slopes of the tangent lines that converge towards the slope at x = a.</p> Signup and view all the answers

<p>The difference quotient represents the slope of the secant line, calculated as $\frac{f(x) - f(a)}{x - a}$.</p> Signup and view all the answers

<p>It denotes taking the limit of the difference quotient as x approaches the point a.</p> Signup and view all the answers

<p>It ensures that the tangent line can be accurately evaluated at that specific point.</p> Signup and view all the answers

<p>The derivative is expressed as $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$.</p> Signup and view all the answers

<p>It implies that as x approaches a, the calculated slopes converge to the exact slope of the tangent line at that point.</p> Signup and view all the answers

<p>The derivative is 𝑓 ′(𝑥) = lim (𝑐 - 𝑐)/ℎ as ℎ approaches 0, which simplifies to 0.</p> Signup and view all the answers

<p>The derivative is calculated as 𝑓 ′(𝑥) = lim (ℎ)/(ℎ) as ℎ approaches 0, resulting in 1.</p> Signup and view all the answers

<p>𝑓 ′(𝑎) = lim (cos(𝑎 + ℎ) - cos(𝑎))/ℎ as ℎ approaches 0 = -sin(𝑎).</p> Signup and view all the answers

<p>The limit process helps determine the instantaneous rate of change, which the derivative represents, by considering values arbitrarily close to a point.</p> Signup and view all the answers

<p>The negative sign indicates that the slope of the cosine function is decreasing, which aligns with its behavior as it transitions from positive to negative values.</p> Signup and view all the answers

A function that reverses the effect of another function. If f(x) = y, then the inverse function, denoted by f⁻¹(y) = x.

Express the input variable (x) in terms of the output variable (y).

Swap the input and output variables (x and y).

A test to determine if a function has an inverse. A function has an inverse if and only if no horizontal line intersects its graph more than once.

A function where f(x1) < f(x2) for all x1 < x2 in its domain.

A function where f(x1) > f(x2) for all x1 < x2 in its domain.

A function has an inverse if and only if it is either strictly increasing or strictly decreasing.

A function where each input (x-value) corresponds to a unique output (y-value).

The formula that expresses the cosine of the difference of two angles (α - β) in terms of the cosines and sines of the individual angles.

Applying the algebraic distributive property to simplify the term [cos (α - β) - 1]² + sin²(α - β) by multiplying and combining like terms.

Applying the algebraic distributive property to simplify the term (cos α - cos β)² + (sin α - sin β)² by multiplying and combining like terms.

Equating the simplified expressions of the left-hand side and right-hand side to arrive at a simplified equation.

The formula that expresses the cosine of the sum of two angles (α + β) in terms of the cosines and sines of the individual angles.

Deriving the cosine sum formula by using the cosine difference formula and substituting (-β) for β.

The process of using previously proven identities and algebraic manipulation to derive a new formula.

An equation that holds true for all possible values of the variables involved, typically involving trigonometric functions.

A type of discontinuity where the function can be made continuous by redefining one point.

If a continuous function on a closed interval takes on two values, it must also take on all values in between.

A function where the graph can be drawn without lifting the pen from the paper.

An interval that includes both its endpoints.

An interval that has finite endpoints.

A value of x where the function equals zero.

The only one of its kind, there can't be another.

The number that, when multiplied by itself n times, equals the given number.

The inverse function of sine (sin) when the domain of sine is restricted to -π/2 ≤ x ≤ π/2.

The inverse function of cosine (cos) when the domain of cosine is restricted to 0 ≤ x ≤ π.

The inverse function of tangent (tan) when the domain of tangent is restricted to -π/2 < x < π/2.

The inverse function of cotangent (cot) when the domain of cotangent is restricted to 0 < x < π.

The arcsin and sin functions are not inverse functions in all cases. They are only inverses when the input of arcsin is within the restricted domain of sine (-π/2 ≤ x ≤ π/2).

The range of arcsin is -π/2 ≤ y ≤ π/2. It's the same as the restricted domain of sine.

The range of arccos is 0 ≤ y ≤ π. It's the same as the restricted domain of cosine.

The range of arctan is -π/2 < y < π/2. It's the same as the restricted domain of tangent.

The derivative of the cosine function, represented by (cos(x))' or d/dx(cos(x)), is equal to -sin(x).

The derivative of the function f(x) = x is 1. This means the slope of the tangent line to the graph of y = x is always 1.

The derivative of a constant function f(x) = c is 0. This indicates that the slope of the tangent line to the graph of y = c is always zero, meaning it's a horizontal line.

The process of determining the derivative of a function involves calculating the instantaneous rate of change of the function at a particular point. This involves finding the limit of the difference quotient as the change in x approaches zero.

The expression representing the average rate of change of a function f(x) over an interval h is called the difference quotient. It is defined as [f(x + h) - f(x)] / h.

A line that touches a curve at a single point and has the same slope as the curve at that point.

The rate of change of a curve at a specific point, represented by the slope of the tangent line at that point.

Taking the limit of the difference quotient as the second point approaches the first point to find the exact slope of the tangent line.

The instantaneous rate of change of a function at a point, obtained by finding the limit of the difference quotient as the second point approaches the first point.

The derivative is used to find the instantaneous rate of change of a function at a given point. It can also be used to find the maximum or minimum points of a function.

A function that assigns a unique output value for every input value.

A point on the graph of the function 𝑓(𝑥) where the input value is 𝑎 and the output value is 𝑓(𝑎).