05 Trigonometric Functions

Measuring Angles

An angle is formed by rotating a ray about its endpoint. The starting ray is the initial side, and the ray after rotation is the terminal side. Counterclockwise rotation is positive, while clockwise rotation is negative.

Degrees and radians

Degrees divide a full revolution into $360^\circ$. A half revolution is $180^\circ$, and a quarter revolution is $90^\circ$.

A radian measures an angle using the ratio of intercepted arc length to radius:

Because a circle has circumference $2\pi r$,

Use

and

For example,

Takeaway: Convert all angles to a consistent unit before evaluating or comparing trigonometric expressions.

The Unit Circle and Exact Values

The unit circle is centered at the origin and has radius $1$:

Starting at $(1,0)$ and rotating counterclockwise through an angle $t$, the terminal point is

Thus, the horizontal coordinate gives cosine and the vertical coordinate gives sine. Substitution into the circle equation gives the Pythagorean identity

The signs depend on the quadrant: both sine and cosine are positive in Quadrant I; sine is positive and cosine is negative in Quadrant II; both are negative in Quadrant III; and cosine is positive while sine is negative in Quadrant IV.

Useful first-quadrant coordinates are

• $0\mapsto(1,0)$

• $\frac{\pi}{6}\mapsto\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)$

• $\frac{\pi}{4}\mapsto\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$

• $\frac{\pi}{3}\mapsto\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$

• $\frac{\pi}{2}\mapsto(0,1)$

The remaining common values follow by symmetry. For an angle outside the first revolution, use periodicity, its quadrant, and its reference angle. For example,

The other basic functions are defined by

Each expression is defined only when its denominator is nonzero.

Takeaway: The unit circle turns angle evaluation into coordinate reading, while the quadrant and reference angle determine the sign.

Graphs and Transformations

The basic functions are $y=\sin x$ and $y=\cos x$. Each has domain $( -\infty,\infty )$, range $[-1,1]$, and period $2\pi$:

Over one period, sine passes through

Cosine passes through

Tangent is

It has period $\pi$, range $( -\infty,\infty )$, and vertical asymptotes where

A transformed sinusoid can be written as

or with cosine in place of sine. Its key features are:

• Amplitude: $|A|$

• Period: $\frac{2\pi}{|B|}$

• Horizontal shift: $h$

• Midline: $y=D$

• Maximum: $D+|A|$

• Minimum: $D-|A|$

For

the amplitude is $3$, the period is $\pi$, the midline is $y=1$, and the graph shifts right by $\frac{\pi}{4}$.

Takeaway: Read a sinusoidal equation by separating vertical size, horizontal repetition, horizontal position, and vertical position.

Identities and Algebraic Relationships

An identity is an equation that is true for every value in the common domain of its expressions.

The reciprocal and quotient identities are

The three principal Pythagorean identity formulas are

Parity identities show that cosine and secant are even, while sine, cosecant, tangent, and cotangent are odd:

• $\cos(-x)=\cos x$ and $\sec(-x)=\sec x$

• $\sin(-x)=-\sin x$, $\csc(-x)=-\csc x$, $\tan(-x)=-\tan x$, and $\cot(-x)=-\cot x$

Sum-and-difference identities include

The double-angle identities include

and

To prove an identity, begin with one side and transform it using known identities until it matches the other side. Monitor excluded values caused by denominators. For example,

Takeaway: Choose identities that simplify the structure, and distinguish an identity that is always true on its domain from an equation whose solutions must be found.

Inverse Trigonometric Functions

The functions $\sin x$, $\cos x$, and $\tan x$ repeat, so they are not one-to-one on their full domains. To define an inverse trigonometric function, restrict the original function to an interval on which it is one-to-one.

The standard restrictions are:

• Sine has restricted domain $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$, so $\arcsin x$ has range $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$.

• Cosine has restricted domain $[0,\pi]$, so $\arccos x$ has range $[0,\pi]$.

• Tangent has restricted domain $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$, so $\arctan x$ has range $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$.

The input restrictions are

while $\arctan x$ accepts every real input. Examples are

The compositions

are valid on the appropriate input domains. The reverse composition must respect the principal range. For example, $\arcsin(\sin x)=x$ only when $x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$.

If $\theta=\arctan x$, a right triangle with opposite side $x$, adjacent side $1$, and hypotenuse $\sqrt{x^2+1}$ gives

Takeaway: Inverse functions return principal angles, not every angle with the same trigonometric value.

Solving Basic Trigonometric Equations

To solve a basic trigonometric equation, first isolate the trigonometric expression. Then find a reference angle or use an inverse function, identify the quadrants where the function has the required sign, and list every solution in the requested interval.

Consider

The reference angle is $\frac{\pi}{6}$. Sine is positive in Quadrants I and II, so

Because sine has period $2\pi$, all real solutions are

Use the relevant period for other functions: sine and cosine repeat every $2\pi$, while tangent repeats every $\pi$. Always check domain restrictions before accepting a result, especially when the equation contains a reciprocal or quotient.

Takeaway: A complete solution requires the correct reference angle and every quadrant or period that satisfies the original equation.