12.2 Trigonometric Ratios and Graphs

12.2 Trigonometric Ratios and Graphs

Quick Answer: Sine, cosine, and tangent are ratios: in a right triangle, opposite/hypotenuse, adjacent/hypotenuse, and opposite/adjacent; in the Cartesian plane, y/r, x/r, and y/x. Special angles 0°, 30°, 45°, 60°, and 90° have exact values you must use by hand. Period, amplitude, domain, and range describe the graph; a, k, and q then stretch, compress, and translate it.

Why trig graphs are a MAT function type

The MAT booklet groups definitions of trigonometric ratios (sine, cosine, tangent) with characteristics of trigonometric functions and their graphs — domain, range, period, amplitude — including transformations of trigonometric functions. Trig graphs sit with the other function types: you read them the same way you read a moved parabola, except the wave repeats. There is no calculator, so every numerical value in this section comes from special angles or from exact algebra. Independent OpenExamPrep examples below are original; they are not official NBT items.

Two definitions you must switch between

Right-triangle definition (acute θ in a right triangle):

• sin θ = opposite / hypotenuse
• cos θ = adjacent / hypotenuse
• tan θ = opposite / adjacent = sin θ / cos θ, provided cos θ ≠ 0

Cartesian definition for an angle in standard position, with a point (x, y) on the terminal ray and r = √(x² + y²) > 0:

• sin θ = y / r
• cos θ = x / r
• tan θ = y / x, provided x ≠ 0

The Cartesian form extends the ratios beyond the first quadrant. Signs follow the CAST pattern:

• Quadrant I (0° to 90°): all positive
• Quadrant II (90° to 180°): sine positive
• Quadrant III (180° to 270°): tangent positive
• Quadrant IV (270° to 360°): cosine positive

Worked example. A point (3, −4) lies on the terminal ray. Then r = 5. So cos θ = 3/5, sin θ = −4/5, and tan θ = −4/3. The angle is in quadrant IV, which matches cosine positive and sine negative. A stem that only gives the point (3, −4) and asks for tan θ is testing the Cartesian definition, not a triangle drawing.

Special angles you compute by hand

MAT expects exact values at 0°, 30°, 45°, 60°, 90° (and the related angles you reach by reduction in the next section). Memorise this table; you will not be entering it into a device.

Two hand checks: sin 30° = cos 60° = 1/2, and tan 45° = 1. tan 90° is undefined because cos 90° = 0. On a graph, that undefined value is a vertical asymptote of y = tan θ.

Domain, range, period, amplitude

Treat each parent as a function of a real angle θ, with degree measure as used throughout NSC-style MAT items.

Period is the smallest positive T such that f(θ + T) = f(θ) for all θ in the domain. Amplitude of a sine or cosine wave is half the distance between the maximum and the minimum: for y = a sin θ it is |a|. Tangent has no amplitude because it has no maximum.

Worked example — reading a described sine graph. A stem says: the graph of y = sin θ is stretched vertically by 3, compressed horizontally so that one full wave fits into 180°, and lifted 1 unit. That is y = 3 sin(2θ) + 1.

• Amplitude 3 → coefficient 3
• Period 180° → 360°/|k| = 180° → |k| = 2
• Vertical shift +1 → plus 1
• Range becomes [−3 + 1, 3 + 1] = [−2, 4]

If the stem instead said the wave was reflected in the x-axis, the equation is y = −3 sin(2θ) + 1. Maxima and minima swap roles relative to the midline y = 1.

Transformations of trig graphs, same language as 12.1

Write y = a sin k(θ − p) + q or y = a cos k(θ − p) + q.

• a: vertical stretch and possible x-axis reflection; amplitude = |a|
• k: horizontal compression; period = 360° / |k| for sine and cosine, and 180° / |k| for tangent
• p: phase shift (right if the form is (θ − p))
• q: vertical shift of the midline to y = q; the range of sine or cosine becomes [q − |a|, q + |a|]

Worked example from turning points described in words. A cosine graph has midline y = −1, maximum 1, and consecutive maxima at θ = 0° and θ = 120°.

• Midline −1 → q = −1
• Maximum 1 is 2 units above the midline → |a| = 2, and because a cosine parent has a maximum at 0°, take a = +2 when a maximum sits at θ = 0°
• Distance between consecutive maxima is the period: 120° → 360°/|k| = 120° → k = 3

So y = 2 cos(3θ) − 1. Check a minimum: cosine minima occur when the inner angle is 180°, so 3θ = 180°, θ = 60°, value = 2(−1) − 1 = −3. The range is [−3, 1].

Worked example — tangent. y = tan(θ − 45°) still has period 180° (k = 1) but its asymptotes move from 90° + 180°k to 135° + 180°k. A stem that lists undefined at 135° and 315° in [0°, 360°) is naming that shift. y = tan(2θ) would instead have period 90° and more asymptotes in the same interval.

Connecting ratios to graphs without a picture

If sin α = 1/2 and α is acute, then α = 30°, and the sine graph at 30° is at height 1/2, heading toward its first maximum at 90°. The cosine graph at 30° is at √3/2. You can answer a which-graph-is-described item from intercepts and one special-angle height: y = sin θ is 0 at 0°, 180°, and 360°; y = cos θ is 1 at 0° and −1 at 180°.

A horizontal shift links the two waves: sin(θ + 90°) = cos θ, so translating the sine graph 90° left produces the cosine graph. Translating sine 90° right produces y = sin(θ − 90°) = −cos θ. Those two identities let you match a described shift to an equation when the stem never prints axes.

MAT tactic for trig graphs

1. Decide sine, cosine, or tangent from intercepts, a maximum at 0° (cosine), or vertical asymptotes (tangent).
2. Read amplitude from (max − min)/2 and midline from (max + min)/2.
3. Read period from consecutive maxima, or from how often asymptotes repeat.
4. Apply special-angle values by hand to test one interior point against the remaining options.
5. Remember the horizontal-shift sign: y = sin(θ − 30°) is 30° right, not left.