Unit 3 Trigonometry Toolkit: Inverses, Identities, and Equivalent Forms

Inverse function

A function that undoes another function’s output; if f has an inverse, then f⁻¹(f(x)) = x (on the appropriate domain).

Horizontal line test

A graph passes this test if every horizontal line intersects it at most once; passing implies the function is one-to-one and therefore invertible (on that domain).

Domain restriction (for invertibility)

Limiting a function’s domain to make it one-to-one so an inverse function can be defined.

Inverse trigonometric function

A function that takes a trig ratio value as input and returns the corresponding principal angle from a restricted range (e.g., arcsin, arccos, arctan).

Arc notation

Notation for inverse trig functions such as arcsin(x), arccos(x), and arctan(x).

Inverse trig notation misconception (sin⁻¹(x))

sin⁻¹(x) means arcsin(x) (inverse sine), not 1/sin(x) (the reciprocal).

arcsin(x)

The angle y in the principal range [−π/2, π/2] such that sin(y) = x (defined only for −1 ≤ x ≤ 1).

arccos(x)

The angle y in the principal range [0, π] such that cos(y) = x (defined only for −1 ≤ x ≤ 1).

arctan(x)

The angle y in the principal range (−π/2, π/2) such that tan(y) = x (defined for all real x).

Principal value

The single “official” output angle chosen from an inverse trig function’s restricted range, ensuring the inverse is a function.

Unit circle coordinate meaning

On the unit circle, cos(θ) is the x-coordinate and sin(θ) is the y-coordinate of the point at angle θ.

Cancellation identity: sin(arcsin(x))

sin(arcsin(x)) = x for −1 ≤ x ≤ 1 (because arcsin outputs angles in sine’s one-to-one interval).

Range-restriction effect: arcsin(sin(θ))

arcsin(sin(θ)) returns the principal angle in [−π/2, π/2] with the same sine value as θ, so it may not equal θ.

Solving trig equations with inverse trig (principal solution)

Using inverse trig to get one principal angle (e.g., θ = arcsin(0.8)), then using symmetry/periodicity to find all solutions on the required interval.

Trigonometric identity

An equation true for all angles in its domain (where both sides are defined), e.g., sin²(θ) + cos²(θ) = 1.

Trigonometric equation

An equation that is true only for specific angle values; the task is to find those values (often on a specified interval).

Reciprocal identities

sec(θ)=1/cos(θ), csc(θ)=1/sin(θ), cot(θ)=1/tan(θ).

Quotient identities

tan(θ)=sin(θ)/cos(θ) and cot(θ)=cos(θ)/sin(θ).

Pythagorean identities

sin²(θ)+cos²(θ)=1; tan²(θ)+1=sec²(θ); 1+cot²(θ)=csc²(θ) (valid where defined).

Even-odd identities

cos is even: cos(−θ)=cos(θ); sin and tan are odd: sin(−θ)=−sin(θ), tan(−θ)=−tan(θ).

Cofunction identities

Relationships for complementary angles: sin(π/2−θ)=cos(θ), cos(π/2−θ)=sin(θ), tan(π/2−θ)=cot(θ).

Extraneous solution (from invalid algebra)

A “solution” created by algebraic steps that change the domain (e.g., multiplying by cos(θ) when cos(θ)=0 is possible); must be rejected by checking in the original equation.

Secant (sec)

The reciprocal of cosine: sec(θ)=1/cos(θ); undefined where cos(θ)=0 (odd multiples of π/2).

Cosecant (csc)

The reciprocal of sine: csc(θ)=1/sin(θ); undefined where sin(θ)=0 (integer multiples of π).

Simplifying sin(arccos(x))

For −1 ≤ x ≤ 1, sin(arccos(x)) = √(1−x²) because arccos(x) outputs θ in [0,π], where sin(θ) ≥ 0.