Trigonometric Function Vocabulary- Key Terms and Definitions
Why This Vocabulary List Exists
Trigonometry has its own language. If you're mixing up amplitude with frequency, or you can't tell your cotangent from your cosecant, this page cuts through the noise. No theory lessons. Just definitions that actually make sense.
The Six Core Trigonometric Functions
Every trig problem starts here. These functions relate angles to side ratios in a right triangle.
Sine (sin)
The ratio of the opposite side to the hypotenuse. Think "sin goes opposite over hypotenuse."
Range: -1 to 1
Cosine (cos)
The ratio of the adjacent side to the hypotenuse. "Cos goes adjacent over hypotenuse."
Range: -1 to 1
Tangent (tan)
The ratio of opposite to adjacent. Also equals sin/cos. This one's undefined at 90° and 270°.
Range: all real numbers
Cosecant (csc)
The reciprocal of sine. Equals hypotenuse/opposite. Undefined where sin = 0.
Range: (-∞, -1] ∪ [1, ∞)
Secant (sec)
The reciprocal of cosine. Equals hypotenuse/adjacent. Undefined where cos = 0.
Range: (-∞, -1] ∪ [1, ∞)
Cotangent (cot)
The reciprocal of tangent. Equals adjacent/opposite. Also equals cos/sin. Undefined where tan = 0.
Range: all real numbers
Graph Properties You Need to Know
Amplitude
Half the distance between the maximum and minimum values of the function. For sin and cos, this is always 1. When you multiply the function by a constant, that constant becomes the new amplitude.
Period
The distance along the x-axis for the function to complete one full cycle. For sin and cos, the period is 2π. For tan and cot, it's π. If you compress or stretch the graph, the period changes.
Phase Shift
Horizontal displacement of the graph. A positive phase shift moves the graph right. A negative one moves it left. This happens when you add or subtract inside the function argument.
Vertical Shift
Movement up or down on the y-axis. This occurs when you add or subtract a constant outside the function.
Frequency
How many cycles occur in a unit interval. Frequency is the reciprocal of period. If period = 2π, frequency = 1/(2π).
Angle Measurement Terms
Degrees
A full circle is 360°. Used in everyday applications and when working with geometry problems.
Radians
A full circle is 2π radians. The natural unit for calculus and higher mathematics. 180° = π radians.
Reference Angle
The acute angle between the terminal side of a given angle and the x-axis. Always between 0 and π/2.
Quadrantal Angles
Angles that land on the x or y axis: 0°, 90°, 180°, 270°, 360°. These angles often produce undefined values for certain functions.
The Unit Circle Essentials
The unit circle has radius 1, centered at the origin. Every point on the circle follows the pattern (cos θ, sin θ).
Key points on the unit circle:
- (1, 0) at 0° or 0 radians
- (0, 1) at 90° or π/2 radians
- (-1, 0) at 180° or π radians
- (0, -1) at 270° or 3π/2 radians
Inverse Trigonometric Functions
These "reverse" the regular trig functions. They take a ratio and return an angle.
- arcsin or sin⁻¹: Returns angle from sine ratio. Domain: [-1, 1]. Range: [-π/2, π/2]
- arccos or cos⁻¹: Returns angle from cosine ratio. Domain: [-1, 1]. Range: [0, π]
- arctan or tan⁻¹: Returns angle from tangent ratio. Domain: all real numbers. Range: (-π/2, π/2)
⚠️ The notation sin⁻¹ does NOT mean 1/sin. That's csc. The -1 is a superscript indicating the inverse function, not an exponent.
Function Comparison Table
| Function | Reciprocal | Domain Restrictions | Range |
|---|---|---|---|
| sin | csc | None | [-1, 1] |
| cos | sec | None | [-1, 1] |
| tan | cot | Odd multiples of π/2 | All real numbers |
| csc | sin | Multiples of π | (-∞, -1] ∪ [1, ∞) |
| sec | cos | Odd multiples of π/2 | (-∞, -1] ∪ [1, ∞) |
| cot | tan | Multiples of π | All real numbers |
Even and Odd Functions
Even functions: cos(-θ) = cos(θ). The graph is symmetric about the y-axis.
Odd functions: sin(-θ) = -sin(θ) and tan(-θ) = -tan(θ). The graph is symmetric about the origin.
Sec, csc, and cot follow the same parity as their reciprocal functions.
Domain and Range at a Glance
| Function | Domain | Range |
|---|---|---|
| sin x | All real numbers | [-1, 1] |
| cos x | All real numbers | [-1, 1] |
| tan x | x ≠ π/2 + nπ | All real numbers |
| csc x | x ≠ nπ | (-∞, -1] ∪ [1, ∞) |
| sec x | x ≠ π/2 + nπ | (-∞, -1] ∪ [1, ∞) |
| cot x | x ≠ nπ | All real numbers |
Getting Started: How to Use This Vocabulary
Step 1: Memorize the three main functions first. Sine, cosine, tangent. Get these down before touching the reciprocals.
Step 2: Learn the unit circle. Memorize the coordinates at 0, π/2, π, and 3π/2. Everything else follows from these.
Step 3: Understand what sine and cosine actually return. They're ratios, not lengths. A sine value of 0.5 doesn't tell you the angle—it tells you the ratio of opposite to hypotenuse.
Step 4: When solving problems, check your domain first. If you're working with tan or sec, verify your angle isn't hitting an asymptote.
Step 5: Use reference angles for quadrant problems. Any angle in Quadrant II, III, or IV can be reduced to its reference angle in Quadrant I.
Common Mistakes to Avoid
- Confusing amplitude with period. Amplitude is vertical height. Period is horizontal width.
- Forgetting that csc, sec, and cot have ranges that exclude values between -1 and 1.
- Using degrees when the problem expects radians, or vice versa.
- Writing sin⁻¹(x) when you mean 1/sin(x). They're completely different operations.
That's the vocabulary. Use it.