Co-function identities: Worked examples — Maths deep dive

This deep-dive drills the parent note Co-function identities (index 3.1.11) into every case a problem can throw at you. We build each symbol from scratch, forecast before solving, and verify every number.

Before anything, the one rule this whole page rests on:

If any of those words feel shaky, revisit Right Triangle Trigonometry (SOH-CAH-TOA) and Complementary and Supplementary Angles first.

The scenario matrix

Here is every class of problem this topic can produce. The worked examples below each carry a cell tag so you can see the whole space is covered.

Cell	What makes it different	Example
A. Plain convert	Angle already looks like $90° - θ$ ; just fire the rule	Ex 1
B. Collapse a sum	Two different angles that are complements; combine with Pythagorean Identities	Ex 2
C. Ratio → 1	Numerator/denominator become identical after a swap	Ex 3
D. Radian form	$\frac{π}{2} - θ$ instead of degrees	Ex 4
E. Sign / wrong-quadrant trap	Angle is $> 90°$ , so it is not a complement — must reroute via Reference Angles and Supplementary Identities	Ex 5
F. Degenerate / boundary	$θ = 0°$ or $θ = 90°$ : check the rule still holds and nothing blows up	Ex 6
G. Chained / nested	Apply the identity twice, or inside another function	Ex 7
H. Real-world word problem	Sun-angle / ramp geometry, needs a figure	Ex 8
I. Exam twist (prove-it)	Prove an expression is constant using the identity + algebra	Ex 9

Everything below fills these nine cells.

Ex 1 — Plain convert (Cell A)

Ex 2 — Collapse a sum (Cell B)

Ex 3 — Ratio collapses to 1 (Cell C)

Ex 4 — Radian form (Cell D)

Ex 5 — The wrong-quadrant trap (Cell E)

Worked example A student writes

$sin 130° = cos (90° - 130°) = cos (- 40°)$ . Is this a valid co-function use? Simplify $sin 130°$ correctly. Forecast: does the co-function rule even apply when the angle is bigger than $90°$ ? Guess yes/no.

Step 1. Check the requirement: co-function pairing needs two angles that sum to exactly $90°$ (both acute). Here $130° > 90°$ , so it has no acute complement — the identity is the wrong tool. Why this step? The whole geometric picture came from a right triangle whose two other angles are acute. Feed in $130°$ and there is no such triangle; blindly writing $90° - 130° = - 40°$ is meaningless as a triangle angle.

Step 2. Use the correct tool instead: $130° = 180° - 50°$ , a supplementary relationship (see Reference Angles and Supplementary Identities). For sine, $sin (180° - θ) = sin θ$ . Why this step? $130°$ lives in Quadrant II where sine is positive; the reference angle is $50°$ , and the supplementary rule (not the co-function rule) governs this case.

Step 3. $sin 130° = sin 50°$ . If you want a co-function form of that, $sin 50° = cos (90° - 50°) = cos 40°$ . Why this step? Now $50°$ is acute, so the co-function rule is legal and turns it into $cos 40°$ .

Answer: $sin 130° = sin 50° = cos 40°$ (and it is positive, $\approx 0.766$ ).

Verify: $sin 130° \approx 0.7660$ , $cos 40° \approx 0.7660$ ✔. The naive " $cos (- 40°)$ " happens to also equal $0.766$ here only because $cos$ is even and $cos (- 40°) = cos 40°$ — a coincidence of sign, not a licence to skip the check.

Ex 6 — Degenerate boundary values (Cell F)

Worked example Test the co-function rule at the edges

$θ = 0°$ and $θ = 90°$ for $tan (90° - θ) = cot θ$ . Where (if anywhere) does something blow up? Forecast: $cot 0°$ — is that a finite number? Predict what breaks.

Step 1. At $θ = 0°$ : left side $tan (90° - 0°) = tan 90°$ , which is undefined (vertical, division by $cos 90° = 0$ ). Right side $cot 0° = \frac{cos 0°}{sin 0°} = \frac{1}{0}$ , also undefined. Why this step? The identity is an equation; for it to be honest, both sides must be undefined at the same inputs. They are — the identity is consistent even at its breakdown point.

Step 2. At $θ = 90°$ : left side $tan (90° - 90°) = tan 0° = 0$ . Right side $cot 90° = \frac{cos 90°}{sin 90°} = \frac{0}{1} = 0$ . Both $0$ ✔. Why this step? This is the opposite extreme; confirming both sides equal $0$ shows the rule holds right up to the boundary.

Step 3. Sanity on the safe interior: at $θ = 30°$ , $tan 60° = 3 \approx 1.732$ and $cot 30° = 3$ ✔.

Answer: The identity holds for all $θ$ where both functions are defined; it is jointly undefined at $θ = 0°$ (where $cot 0°$ and $tan 90°$ both explode). No contradiction.

Verify: $tan 60° = 3$ and $cot 30° = 3$ , so their difference is $0$ ✔.

Ex 7 — Chained / nested application (Cell G)

Ex 8 — Real-world word problem (Cell H)

Worked example A straight ramp leans against a wall. The angle the ramp makes with the

ground is $θ = 35°$ . What angle does it make with the wall, and how does the ramp's "sine seen from the ground" relate to its "cosine seen from the wall"? Forecast: the ground and the wall meet at a right angle. What must the two ramp-angles add to?

Step 1. Ground ⟂ wall means the corner is $90°$ . The ramp, ground, and wall form a right triangle, so the angle at the wall is $ϕ = 90° - 35° = 55°$ (look at the figure: the two coloured angles fill the square corner). Why this step? The two non-right angles of any right triangle are complementary — the geometric backbone of every co-function identity.

Step 2. From the ground's viewpoint, the ramp's height $=$ opposite, so $sin 35° = \frac{height}{ramp}$ . From the wall's viewpoint, that same height is adjacent, so $cos 55° = \frac{height}{ramp}$ . Why this step? One physical side is "opposite" to one observer and "adjacent" to the other — this is why $sin 35° = cos 55°$ .

Step 3. Therefore $sin 35° = cos (90° - 35°) = cos 55°$ .

Answer: wall angle $= 55°$ , and $sin 35° = cos 55°$ .

Verify: $sin 35° \approx 0.5736$ and $cos 55° \approx 0.5736$ ✔; angles $35° + 55° = 90°$ ✔.

Ex 9 — Exam twist: prove a constant (Cell I)

Recall Self-test: match each problem to its matrix cell

$\frac{cos 10°}{sin 80°}$ is which cell? ::: Cell C (ratio → 1). $sin 145°$ is which cell, and which tool? ::: Cell E — supplementary/reference, NOT co-function. $tan 90°$ vs $cot 0°$ is which cell? ::: Cell F (degenerate boundary; both undefined). $cos^{2} 18° + cos^{2} 72°$ is which cell? ::: Cell B (collapse a sum) → equals $1$ .

Connections

Parent: Co-function identities (index 3.1.11)
Right Triangle Trigonometry (SOH-CAH-TOA) — opp/adj/hyp definitions used in Ex 8.
Pythagorean Identities — the collapse in Ex 2.
Reference Angles and Supplementary Identities — the correct route for Ex 5.
Complementary and Supplementary Angles — why the ramp angles add to $90°$ .
Radian Measure — the $π /2$ form in Ex 4.
Angle Sum and Difference Formulas — the general parent of these identities.