Understand the three fundamental trig functions through definitions, graphs, properties, and transformations
There are two complementary definitions for the three primary trigonometric functions. Both are correct, and understanding both gives you a complete picture of what these functions truly represent.
In a right triangle with an acute angle θ, the three functions are defined as ratios of the triangle's sides:
This definition works perfectly for angles between 0° and 90°, because those are the only angles that fit inside a right triangle. It is the foundation of SOHCAHTOA and is the most intuitive starting point for beginners.
For any angle θ (including angles larger than 90° or negative angles), draw the angle in standard position on the unit circle. The terminal side intersects the unit circle at some point (x, y). Then:
This definition extends trigonometry to all real numbers and is the one used in calculus, physics, and engineering. The right-triangle definition is actually a special case of the unit circle definition for first-quadrant angles.
The right-triangle definition is great for solving triangle problems. The unit circle definition is essential for understanding trig functions as functions of all real numbers, their graphs, and their periodic behavior.
Understanding the domain and range of each function is critical for graphing and for solving equations correctly.
| Function | Domain | Range | Period |
|---|---|---|---|
| sin θ | All real numbers | [-1, 1] | 2π (360°) |
| cos θ | All real numbers | [-1, 1] | 2π (360°) |
| tan θ | All reals except θ = π/2 + nπ | (-∞, +∞) | π (180°) |
Sine and cosine are defined for every real number and their outputs are always between −1 and 1, inclusive. They achieve the value 1 and −1 at their peaks and valleys.
Tangent is undefined wherever cos(θ) = 0, which occurs at θ = 90°, 270°, −90°, etc. (or θ = π/2 + nπ in radians). At these points, the tangent function has vertical asymptotes. Between the asymptotes, tangent takes on all real values from −∞ to +∞.
All three trig functions are periodic, meaning they repeat their values at regular intervals:
Sine and cosine repeat every 2π radians (360°). Tangent repeats every π radians (180°), making it the fastest-repeating of the three. This periodic nature reflects the fact that going around the unit circle brings you back to the same point.
A function is even if f(−x) = f(x) and odd if f(−x) = −f(x). For the trig functions:
The most fundamental trig identity connects sine and cosine:
This identity comes directly from the unit circle equation x² + y² = 1. It is true for every value of θ and is the basis for deriving dozens of other identities.
The sine function starts at 0, rises to 1 at π/2, returns to 0 at π, drops to −1 at 3π/2, and returns to 0 at 2π. It is a smooth, continuous wave that repeats every 2π.
The cosine function starts at 1, drops to 0 at π/2, reaches −1 at π, returns to 0 at 3π/2, and comes back to 1 at 2π. It has the same shape as the sine wave but is shifted π/2 to the left. In fact, cos(θ) = sin(θ + π/2).
The tangent graph looks very different from sine and cosine. It has vertical asymptotes at every odd multiple of π/2, where the function is undefined. Between asymptotes, it sweeps from −∞ through 0 to +∞. It repeats every π radians (180°), half the period of sine and cosine.
The general form of a sinusoidal function is:
Each parameter controls a different transformation:
Amplitude = |A|, Period = 2π/|B|, Phase shift = C/B, Vertical shift = D. With these four parameters, you can describe any sinusoidal wave.
Sine, cosine, and tangent are deeply interconnected:
This relationship explains why tangent is undefined when cos(θ) = 0: you cannot divide by zero. It also explains tangent's sign pattern: tangent is positive when sine and cosine have the same sign (Quadrants I and III) and negative when they have different signs (Quadrants II and IV).
Additionally, cosine is a shifted version of sine:
These are called cofunction identities and reflect the fact that sine and cosine are complementary: the sine of an angle equals the cosine of its complement.
Each of the three primary trig functions has a reciprocal function:
Cosecant (csc) is the reciprocal of sine, secant (sec) is the reciprocal of cosine, and cotangent (cot) is the reciprocal of tangent. These functions are undefined wherever their counterparts equal zero. Together, all six functions form a complete system for describing angles and triangles.
Step 1: Recall that π/4 radians = 45°.
Step 2: On the unit circle, the point at 45° is (√2/2, √2/2).
Step 3: sin(θ) = y-coordinate = √2/2.
Answer: sin(π/4) = √2/2 ≈ 0.7071
Step 1: Compare with y = A cos(Bx − C) + D. Here A = 2, B = 1, C = 0, D = −1.
Step 2: Amplitude = |A| = 2. The wave reaches 2 units above and below the midline.
Step 3: Period = 2π/|B| = 2π/1 = 2π. One complete cycle takes 2π.
Step 4: Phase shift = C/B = 0. No horizontal shift.
Step 5: Vertical shift = D = −1. The midline is y = −1.
Answer: The graph is a cosine wave with amplitude 2, period 2π, midline y = −1, oscillating between y = 1 and y = −3.
Step 1: tan(x) = 1 when the y-coordinate and x-coordinate on the unit circle are equal (since tan = y/x).
Step 2: In the first quadrant, this happens at x = π/4 (45°), where the point is (√2/2, √2/2).
Step 3: Tangent is also positive in Quadrant III. The corresponding angle is π/4 + π = 5π/4 (225°).
Answer: x = π/4 and x = 5π/4
Evaluate cos(5π/6) exactly using the unit circle.
5π/6 = 150°, which is in Quadrant II. Reference angle = 180° − 150° = 30°. cos(30°) = √3/2. In Quadrant II, cosine is negative. Therefore cos(5π/6) = −√3/2.
For y = −3 sin(2x + π) + 4, find the amplitude, period, phase shift, and vertical shift.
Rewrite as y = −3 sin(2(x + π/2)) + 4. Now A = −3, B = 2, C = −π, D = 4.
Amplitude = |A| = 3. Period = 2π/|B| = 2π/2 = π. Phase shift = C/B = −π/2 (shift π/2 to the left). Vertical shift = 4 (up). The graph is also reflected over the midline because A is negative.
The graph of y = sin(x) passes through the point (π/6, 0.5). Verify this and find another point where sin(x) = 0.5 in [0, 2π).
sin(π/6) = sin(30°) = 1/2 = 0.5. Verified. Sine is also 0.5 in Quadrant II where the reference angle is π/6. That angle is π − π/6 = 5π/6. So sin(5π/6) = 0.5.
Sine, cosine, and tangent are the three pillars of trigonometry. They are defined both as triangle ratios and as unit circle coordinates. Their graphs are periodic waves whose shape can be fully controlled by four parameters: amplitude, period, phase shift, and vertical shift.