Trigonometric Functions of Any Angle: Definitions, Properties, and Applications

Trigonometric Functions of Any Angle

Definition and General Formulas

Trigonometric functions can be defined for any angle, not just acute angles in right triangles. For an angle θ in standard position, and a point (a, b) (other than the origin) on its terminal side, the six trigonometric functions are defined as follows, where r is the distance from the origin to (a, b):

• sin θ = b / r

• cos θ = a / r

• tan θ = b / a

• csc θ = r / b

• sec θ = r / a

• cot θ = a / b

where $r = \sqrt{a^2 + b^2}$ and denominators must not be zero.

Example: Finding Trigonometric Functions from a Point

Given a point (12, –5) on the terminal side of angle θ, find the exact values of all six trigonometric functions:

• Calculate $r = \sqrt{12^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13$

• $\sin \theta = \frac{-5}{13}$

• $\cos \theta = \frac{12}{13}$

• $\tan \theta = \frac{-5}{12}$

• $\csc \theta = \frac{13}{-5}$

• $\sec \theta = \frac{13}{12}$

• $\cot \theta = \frac{12}{-5}$

Coterminal Angles

Definition and Properties

Coterminal angles are angles in standard position that share the same terminal side. Two angles θ and φ are coterminal if their difference is a multiple of 360° (or 2π radians):

• $\theta$ and $\theta + 360^\circ k$ (for integer k) are coterminal.

Examples: Finding Trigonometric Values Using Coterminal Angles

• To find $\cos 390^\circ$, subtract 360°: $390^\circ - 360^\circ = 30^\circ$. So, $\cos 390^\circ = \cos 30^\circ$.

• To find $\cos 420^\circ$, subtract 360°: $420^\circ - 360^\circ = 60^\circ$. So, $\cos 420^\circ = \cos 60^\circ$.

• For negative angles, add 360° (or 2π): $-270^\circ + 360^\circ = 90^\circ$. So, $\csc(-270^\circ) = \csc(90^\circ)$.

• For radian measures, add or subtract $2\pi$ as needed. For example, $\frac{9\pi}{4} - 2\pi = \frac{9\pi}{4} - \frac{8\pi}{4} = \frac{\pi}{4}$, so $\cos\left(\frac{9\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right)$.

Signs of Trigonometric Functions in Quadrants

Determining Signs by Quadrant

The sign of each trigonometric function depends on the quadrant in which the terminal side of the angle lies:

Quadrant of θ	sin θ, csc θ	cos θ, sec θ	tan θ, cot θ
I	Positive	Positive	Positive
II	Positive	Negative	Negative
III	Negative	Negative	Positive
IV	Negative	Positive	Negative

Example: Determining the Quadrant

If $\sin \theta < 0$ and $\cos \theta > 0$, then θ is in quadrant IV, since only in quadrant IV is sine negative and cosine positive.

Reference Angles

Definition and Properties

A reference angle is the acute angle formed by the terminal side of θ and the x-axis. It is always between 0° and 90° (or 0 and $\frac{\pi}{2}$ radians).

Finding Reference Angles

• Quadrant I: Reference angle is θ itself.

• Quadrant II: Reference angle is $180^\circ - \theta$ or $\pi - \theta$.

• Quadrant III: Reference angle is $\theta - 180^\circ$ or $\theta - \pi$.

• Quadrant IV: Reference angle is $360^\circ - \theta$ or $2\pi - \theta$.

Examples: Reference Angles

• Reference angle for 150° is 30°.

• Reference angle for –45° is 45°.

• Reference angle for $\frac{9\pi}{4}$ is $\frac{\pi}{4}$.

• Reference angle for $-\frac{5\pi}{6}$ is $\frac{\pi}{6}$.

Using Reference Angles to Find Trigonometric Values

The value of a trigonometric function for any angle θ is equal to the value of the function at its reference angle, up to a sign determined by the quadrant:

• $\sin \theta = \pm \sin \alpha$

• $\cos \theta = \pm \cos \alpha$

• $\tan \theta = \pm \tan \alpha$

The sign depends on the quadrant in which θ lies.

Examples: Using Reference Angles

• $\cos 135^\circ$: Reference angle is 45°, and cosine is negative in quadrant II, so $\cos 135^\circ = -\frac{\sqrt{2}}{2}$.

• $\tan 600^\circ$: Reference angle is 60°, and tangent is positive in quadrant III, so $\tan 600^\circ = \sqrt{3}$.

Summary: Steps for Finding Trigonometric Values of Any Angle

1. If the angle is a quadrantal angle (multiple of 90°), use the definition directly.

2. If the angle is not quadrantal:

   1. Find the reference angle α.

   2. Find the value of the trigonometric function at α.

   3. Adjust the sign based on the quadrant.

Application: Finding All Trigonometric Functions Given One Value

If you know one trigonometric function and the quadrant, you can find all others using the Pythagorean identity and sign rules. For example, if $\tan \theta = -3$ and $\cos \theta < 0$, θ is in quadrant II. Construct a right triangle with sides 1 and 3, hypotenuse $\sqrt{10}$, and assign signs according to the quadrant.

Additional info: The above notes are based on standard precalculus curriculum and include all major concepts from the provided materials, with relevant images included only where they directly support the explanation.