Trigonometry Functions and Inverses Exam

The inverse trigonometric functions are functions that are the inverse of the trigonometric functions.

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

The range of a function is the set of all possible output values (y-values) that the function can produce.

The principal value of an inverse trigonometric function is the specific value of the function that lies within a defined interval. For instance, the principal value of
[cos^ − 1^(−\frac{1}{2})]{.math.inline} lies within [0, π]{.math.inline}, which is [\frac{2π}{3}]{.math.inline}.

The value of a trigonometric function inside an inverse trigonometric function is simplified to the angle itself where the function is defined. For instance, the value of
[sin^ − 1^(sin \frac{3π}{5})]{.math.inline} is [\frac{2π}{5}]{.math.inline} because [sin \frac{3π}{5} = sin \frac{2π}{5}]{.math.inline}

Inverse trigonometric functions are restricted to specific ranges to ensure they have a unique output for every input. If a function is not restricted, it wouldn't have a unique inverse function.

The domain of [arcsin x]{.math.inline} is [-1, 1] because the sine function only outputs values between -1 and 1.

The range of [arcsin x]{.math.inline} is [-\frac{π}{2}, \frac{π}{2}] because it's restricted to the principal values in the interval.

To express an inverse trigonometric function in terms of the other, the given equation's cosine value should be found by applying the Pythagorean identity. For instance, given [cos^ − 1^\frac{x}{2}=0]{.math.inline}, using the Pythagorean identity [sin^2θ + cos^2θ = 1]{.math.inline}, we get [sin^2θ + (\frac{x}{2})^2 = 1]{.math.inline}, simplifying to [sin^2θ = 1 - (\frac{x}{2})^2]{.math.inline} and [sinθ = \sqrt{1-(\frac{x}{2})^2}]{.math.inline}, yielding [x = 1]{.math.inline}.

The value of a trigonometric function inside an inverse trigonometric function is simplified to the corresponding angle. For instance, the value of [cos(sin^ − 1^\frac{1}{2})]{.math.inline} is [\frac{√3}{2}]{.math.inline} because when [sinθ = \frac{1}{2}]{.math.inline}, [cosθ = \frac{√3}{2}]{.math.inline}.

The set of angles that satisfy the equation [arcsin (-1)]{.math.inline} can be expressed as [-\frac{π}{2} + 2*nπ]{.math.inline} because the sine function equals -1 at the angle [-\frac{π}{2}]{.math.inline} and its multiples.

The value of a trigonometric function inside an inverse trigonometric function can often be simplified by visualizing the triangle formed by the input angle and the trigonometric value. For instance, finding the value of [cos(sin^ − 1^\frac{3}{4} + cos^ − 1^\frac{5}{13})]{.math.inline} involves visualizing the triangles and applying the cosine rule. This could be simplified further by understanding the relationship between sine and cosine in the respective triangles and making use of the angles between the two triangles.

The value of [cot(sin^ − 1^x)]{.math.inline} is found by using the trigonometric identity [cotθ = \frac{cosθ}{sinθ}]{.math.inline} and applying the Pythagorean identity. This is solved by considering the angle in the sin^ − 1^x and simplifying.

The value of [θ = arctan(cot \frac{π}{4})]{.math.inline} is [\frac{π}{4}]{.math.inline} because [cot \frac{π}{4}=1]{.math.inline}, and [arctan 1 = \frac{π}{4}]{.math.inline}.

Inverse trigonometric functions are not defined for all values. For instance, [arcsin (-2)]{.math.inline} is not defined because the sine function doesn't output values outside of the range [-1, 1]. Hence, the inverse sine function is not defined for values outside of that range.

Trigonometric identities can be used to simplify compound angle expressions involving inverse trigonometric functions. For example, [cos(2arctan \frac{4}{3})]{.math.inline} can be simplified using the double angle formula for cosine and understanding that [arctan \frac{4}{3}]{.math.inline} represents an angle in a right triangle.

The tangent of a double angle can be calculated using the tangent double angle formula. For example, the tangent of [2sin^ − 1^\frac{√3}{2}]{.math.inline} can be found using the formula [tan 2θ = \frac{2 tanθ}{1 - tan^2θ}]{.math.inline} and the knowledge of [sin^ − 1^\frac{√3}{2}]{.math.inline}.

The value of [sec^ − 1^ (- \frac{2√3}{3})]{.math.inline} can be found by simplifying the argument and applying the definition of secant and arccos. This means [sec^ − 1^ (- \frac{2√3}{3})]{.math.inline} is equivalent to [cos^ − 1^ ( - \frac{√3}{2})]{.math.inline}.

The range of the arctangent function spans all the real numbers.

The principal value branch of [sec^ − 1^x]{.math.inline} is [0 ≤ θ ≤ π, θ ≠ \frac{π}{2}]{.math.inline}.

The range of [arctan 2x]{.math.inline} is the same as the range of the arctangent function, which is [-\frac{π}{2}, \frac{π}{2}]{.math.inline}.

The solution of [tan^ − 1^x - tan^ − 1^y = tan^ − 1^A]{.math.inline} is [A = \frac{x - y}{1 + xy}]{.math.inline}. This formula is derived from the tangent of a difference formula for angles.

The solution to the equation [tan^-1(x + \frac{√2}{2}) = \frac{π}{4}]{.math.inline} can be found using the fact that the tangent of [\frac{π}{4}]{.math.inline} is 1.

Polar coordinates represent a point in a plane using a distance from the origin (r) and an angle from the positive x-axis (θ).

The polar coordinate representation of a point depends on the angle from the positive x-axis and the length of the line connecting the point to the origin (radius).

The polar coordinate representation of a point can have multiple forms. For instance, the point [(5, \frac{2π}{3})]{.math.inline} can also be represented as [(5, \frac{8π}{3})]{.math.inline}.

For any given point in polar coordinates, there are infinite representations by adding 2πn to the angle, where n is an integer.

The point represented by the polar coordinates [(6.5, 300°)]{.math.inline} is located 6.5 units away from the origin at an angle of 300° from the positive x-axis.

The point represented by the polar coordinates [(7.8, 0°)]{.math.inline} is located 7.8 units away from the origin at an angle of 0° from the positive x-axis.