7: Trigonometric Foundations

Angles and Their Measures

Trigonometry connects an angle with a numerical ratio or function value. These connections support calculations involving triangles and models of repeating behavior such as tides, sound waves, rotating machinery, and seasonal data.

An angle is formed by two rays with a common endpoint, called the vertex. The initial ray is the starting side, and the terminal ray is the ending side. In standard position, the vertex is at the origin and the initial side lies on the positive $x$-axis.

Angles can be measured in degrees or radians. A positive angle rotates counterclockwise, while a negative angle rotates clockwise. One complete counterclockwise rotation is $360^\circ$, or $2\pi$ radians, and a half rotation is $180^\circ$, or $\pi$ radians.

A radian is defined from arc length. If an angle intercepts an arc of length $s$ on a circle of radius $r$, then

For a unit circle, $r=1$, so the radian measure equals the intercepted arc length.

To convert degrees to radians, multiply by $\frac{\pi}{180^\circ}$. To convert radians to degrees, multiply by $\frac{180^\circ}{\pi}$. For example,

and

Coterminal angles have the same initial and terminal sides. They can be generated from an angle $\theta$ by adding $360^\circ k$ or $2\pi k$, where $k$ is an integer. Thus, $\frac{\pi}{3}$, $\frac{7\pi}{3}$, and $-\frac{5\pi}{3}$ are coterminal.

Takeaway: Use the conversion factors based on $180^\circ=\pi$ radians, and use coterminal angles to replace difficult angles with equivalent, easier ones.

The Unit Circle and Reference Angles

The unit circle is centered at $(0,0)$, has radius $1$, and satisfies

When an angle $\theta$ is placed in standard position, its terminal side meets the unit circle at a point $(x,y)$. The circular definitions are

Therefore, every point on the unit circle can be written as

Substituting these coordinates into $x^2+y^2=1$ produces the Pythagorean identity:

The quadrant determines the signs of the coordinates:

• In Quadrant I, sine and cosine are positive.

• In Quadrant II, sine is positive and cosine is negative.

• In Quadrant III, sine and cosine are negative.

• In Quadrant IV, sine is negative and cosine is positive.

Because $\tan\theta=\frac{\sin\theta}{\cos\theta}$, tangent is positive in Quadrants I and III and negative in Quadrants II and IV.

A reference angle is the acute positive angle between the terminal side and the $x$-axis. For angles between $0$ and $2\pi$, its formula is

• Quadrant I: $\alpha=\theta$

• Quadrant II: $\alpha=\pi-\theta$

• Quadrant III: $\alpha=\theta-\pi$

• Quadrant IV: $\alpha=2\pi-\theta$

For example, the reference angle for $\frac{5\pi}{6}$ is

Since $\frac{5\pi}{6}$ lies in Quadrant II, sine is positive, so

Takeaway: The unit-circle coordinates provide both the value and the sign of sine and cosine; the reference angle supplies a familiar magnitude.

The Six Trigonometric Functions

For an acute angle in a right triangle, the six trigonometric functions are defined by side ratios:

The reciprocal functions are

The unit-circle definitions of sine and cosine extend these functions to all real angles, including negative angles and angles greater than one complete revolution. The quotient relationships are

A function is undefined when its denominator is zero. Consequently:

• $\tan\theta$ and $\sec\theta$ are undefined when $\cos\theta=0$.

• $\cot\theta$ and $\csc\theta$ are undefined when $\sin\theta=0$.

The reciprocal identities and quotient identities allow the same relationship to be written in different forms, which is useful when simplifying expressions or determining an unknown value.

Takeaway: Sine and cosine describe unit-circle coordinates, while tangent, cotangent, secant, and cosecant are built from ratios or reciprocals of those primary functions.

Special-Angle Values

Special-angle values come from the $30^\circ$-$60^\circ$-$90^\circ$ and $45^\circ$-$45^\circ$-$90^\circ$ triangles. The first-quadrant values are:

• At $0$: $\cos\theta=1$, $\sin\theta=0$, and $\tan\theta=0$.

• At $\frac{\pi}{6}$, or $30^\circ$: $\cos\theta=\frac{\sqrt{3}}{2}$, $\sin\theta=\frac{1}{2}$, and $\tan\theta=\frac{\sqrt{3}}{3}$.

• At $\frac{\pi}{4}$, or $45^\circ$: $\cos\theta=\frac{\sqrt{2}}{2}$, $\sin\theta=\frac{\sqrt{2}}{2}$, and $\tan\theta=1$.

• At $\frac{\pi}{3}$, or $60^\circ$: $\cos\theta=\frac{1}{2}$, $\sin\theta=\frac{\sqrt{3}}{2}$, and $\tan\theta=\sqrt{3}$.

• At $\frac{\pi}{2}$, or $90^\circ$: $\cos\theta=0$, $\sin\theta=1$, and tangent is undefined.

A useful memory pattern for first-quadrant sine values at $0$, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, and $\frac{\pi}{2}$ is

The corresponding cosine values appear in reverse order. To use these values in another quadrant, keep the reference-angle magnitude and apply the sign dictated by the quadrant. For example,

and

Takeaway: Memorize the first-quadrant values, then use reflection through the unit circle and quadrant signs to evaluate the remaining standard angles.

Fundamental Trigonometric Identities

An identity is an equation that is true for every value in the domain of the expressions involved. Identities let you rewrite, simplify, and verify trigonometric expressions.

The reciprocal identities are

The quotient identities are

Starting with $\sin^2\theta+\cos^2\theta=1$, divide by $\cos^2\theta$ or $\sin^2\theta$ to obtain the other Pythagorean identities:

Sine and tangent are odd functions, while cosine is even:

The cofunction identities describe complementary angles:

For example, if $\sin\theta=\frac{3}{5}$ and $\theta$ lies in Quadrant II, then

so

Thus, $\cos\theta=\pm\frac{4}{5}$, and Quadrant II determines the negative choice:

Takeaway: Choose an identity that matches the information given, and always use the domain and quadrant to select the valid sign.

A Systematic Evaluation Strategy

A reliable evaluation procedure combines angle measure, reference angles, special values, and quadrant signs:

1. Express the angle in degrees or radians as required.

2. If useful, replace the angle with a coterminal angle between $0$ and $2\pi$.

3. Identify the quadrant and calculate the reference angle.

4. Use the special-angle value for the reference angle.

5. Apply the sign determined by the quadrant.

6. Check the result with the unit circle or a relevant identity.

For example, evaluate $\tan\left(\frac{11\pi}{6}\right)$. The angle lies in Quadrant IV, so its reference angle is

Tangent is negative in Quadrant IV. Therefore,

The same workflow applies to negative angles and angles larger than $2\pi$: first reduce the angle to an equivalent position, then use the reference angle and quadrant.

These foundations make trigonometric functions useful for describing periodic behavior. A repeating phenomenon can be represented by functions whose values cycle as the input angle changes, with the unit circle and trigonometric identities providing the essential tools for evaluating and transforming those functions.

Final takeaway: Convert carefully, reduce when helpful, identify the reference angle, apply the quadrant sign, and verify the result with an identity or a unit-circle coordinate.