IB Maths AA SLTopic 3 — Trig Equations & IdentitiesPaper 1 & 2~10 min read
Relationship Between Trigonometric Ratios
Sin, cos, and tan aren’t three separate things — they’re best friends who tell on each other. If you know one, you can always find the other two. This note shows you exactly how, and how to pick the right sign every time.
📘 What you need to know
Sin and cos swap roles when angles add to 90°: sin θ = cos(90° − θ) and cos θ = sin(90° − θ).
If you know one trig ratio, you can find the other two using either a right-angled triangle or the identities.
The Pythagorean identity gives you the missing ratio: sin2θ + cos2θ = 1.
The tan identity links them: tan θ = sin θcos θ.
Always check the quadrant to decide if your answer should be positive or negative.
Sin and cos are linked through 90°
Pick any angle θ. The sin of that angle is exactly the same as the cos of (90° − θ). And the cos of θ equals the sin of (90° − θ). They literally swap places.
Complementary identities
sin θ = cos(90° − θ)
cos θ = sin(90° − θ)
Why? Look at any right-angled triangle. The two non-right angles must add up to 90°. If one is θ, the other is (90° − θ). And what’s the “opposite” side from one angle’s view is the “adjacent” side from the other angle’s view — so sin and cos just trade places.
The two angles in a right-triangle add to 90°
“Complementary” just means the two angles add up to 90°. Once you see this swap visually, you’ll never forget it — sin and cos are mirror twins of each other across 45°.
Finding one trig ratio from another
You’ll often be given one ratio (say, sin θ = 3/5) and asked to find another (cos θ or tan θ). There are two equally good ways to do this — pick whichever you find easier.
Method 1
Right-angled triangle + SOHCAHTOA
Draw a right-angled triangle with angle θ.
Label the sides using the ratio you’re given. e.g. sin = 3/5 → opposite = 3, hypotenuse = 5.
Find the missing side using Pythagoras.
Read off the ratio you want using SOHCAHTOA.
Check the sign using the quadrant.
Method 2
Use the identities directly
Start with sin2θ + cos2θ = 1.
Substitute the ratio you know.
Solve for the missing squared ratio.
Square root — remember the ± sign.
Check the sign using the quadrant, then if you also need tan, use tan = sin/cos.
Method 1 (the triangle) is great when the angle is acute. Method 2 (identities) is cleaner when the angle is in a different quadrant — because the ± sign reminds you to think about the quadrant.
Picking the right sign — the four quadrants
Once you have a value like cos θ = ± 4/5, you need to decide whether the answer is positive or negative. The size of the angle tells you which “quadrant” it lives in — and each quadrant has its own sign rules.
Which trig ratios are POSITIVE in each quadrant?
Q2
90° < θ < 180°
Only sin is +
cos & tan are −
Q1
0° < θ < 90°
All are +
sin, cos, tan all positive
Q3
180° < θ < 270°
Only tan is +
sin & cos are −
Q4
270° < θ < 360°
Only cos is +
sin & tan are −
🧠
Memory trick: “All Students Take Calculus”
Starting in Q1 and going anti-clockwise: All positive → Sin positive → Tan positive → Cos positive. Whatever letter is in that quadrant, that’s the only ratio that’s positive there. Everything else is negative.
Sign table — the same info, table form
Q1 0°–90°
Q2 90°–180°
Q3 180°–270°
Q4 270°–360°
sin θ
+
+
−
−
cos θ
+
−
−
+
tan θ
+
−
+
−
📍
Radians? Same idea, different numbers
If the question uses radians instead of degrees, the quadrants are: Q1: 0 to π/2 | Q2: π/2 to π | Q3: π to 3π/2 | Q4: 3π/2 to 2π. The signs are exactly the same.
Worked examples
WE 1
Use the sin–cos complementary identity
Without using a calculator, find the exact value of cos 60° given that sin 30° = 12.
cos 60° = ?30° and 60° add to 90°, so they’re complementary — sin and cos swap.Use the identitycos θ = sin(90° − θ)Let θ = 60°:cos 60° = sin(90° − 60°) = sin 30°Substitute:cos 60° = 12cos 60° = 12
WE 2
Find cos θ and tan θ using a right triangle (acute angle)
Given that sin θ = 35 and θ is acute, find the exact values of cos θ and tan θ.
sin θ = 35, θ acuteθ acute means Q1 — all ratios will be positive. Draw the triangle.Label sides:opposite = 3, hypotenuse = 5Find adjacent (Pythagoras):adj = √(52 − 32) = √16 = 4Read off cos:cos θ = adjhyp = 45Read off tan:tan θ = oppadj = 34cos θ = 45, tan θ = 34acute angle = Q1 = all positive, so no sign issues here
WE 3
Find cos α when the angle is in Q2
The value of sin α = 35 for π2 ≤ α ≤ π. Find the exact value of cos α.
sin α = 35, π2 ≤ α ≤ πα is in Q2 (between π/2 and π) — so cos will be negative.Use the identitysin2 α + cos2 α = 1Substitute:(35)2 + cos2 α = 1Simplify:925 + cos2 α = 1Rearrange:cos2 α = 1625Square root:cos α = ± 45Q2 → cos negative, so:cos α = − 45if you forget the quadrant check, you’ll get the wrong sign every time!
WE 4
Find sin θ and tan θ when θ is in Q3
Given that cos θ = − 513 and 180° < θ < 270°, find the exact values of sin θ and tan θ.
cos θ = − 513, θ in Q3Q3 → only tan is positive, so sin will be negative and tan will be positive.Use the identitysin2 θ + cos2 θ = 1Substitute:sin2 θ + (− 513)2 = 1Simplify:sin2 θ + 25169 = 1Rearrange:sin2 θ = 144169Square root:sin θ = ± 1213Q3 → sin negative:sin θ = − 1213Find tan = sin/cos:tan θ = −12/13−5/13 = 125sin θ = − 1213, tan θ = 125two negatives divided = positive, which matches “tan is + in Q3” ✓
WE 5
Find sin 2α and cos 2α when α is in Q2
The value of sin α = 35 for π2 ≤ α ≤ π. Find the exact values of:
(a) sin 2α(b) cos 2α
From WE 3 we already know cos α = −4/5 (Q2 → cos negative). Now use the double angle formulas.part (a)Use sin 2α = 2 sin α cos α:sin 2α = 2 · 35 · (−45) = −2425sin 2α = − 2425part (b)Use cos 2α = cos2 α − sin2 α:cos 2α = (−45)2 − (35)2 = 1625 − 925 = 725cos 2α = 725finding cos α first unlocks all the double angle answers — that’s the typical exam pattern
💡 Top tips
Always check the quadrant first. Before you do any algebra, work out which quadrant the angle is in — that tells you the sign of every answer.
Method 1 (triangle) is fastest for acute angles. Just draw, label, Pythagoras, read off. No identity needed.
Method 2 (identities) is best for non-acute angles. The ± from the square root forces you to think about the sign — which is what you want.
Remember “All Students Take Calculus.” Q1: All. Q2: Sin. Q3: Tan. Q4: Cos. Whatever’s named is the only one that’s positive in that quadrant.
If you’re given sin and need tan, the fastest route is: find cos using sin² + cos² = 1, then divide tan = sin/cos.
Watch the radians. π/2 ≈ 1.57, π ≈ 3.14, 3π/2 ≈ 4.71, 2π ≈ 6.28. Same quadrants, different numbers.
For complementary identities, the quickest check is: does sin θ + cos(90° − θ) make sense as the same thing? Yes — they’re equal.
⚠ Common mistakes
Forgetting to check the quadrant. If the question says “θ is in Q3” and you write cos θ = +4/5, you’ve lost marks — Q3 means cos is negative.
Forgetting the ± when square rooting. cos² θ = 16/25 doesn’t immediately give cos θ = 4/5 — it gives cos θ = ± 4/5, then you pick the sign.
Mixing up sin and cos in the complementary identity. sin θ = cos(90 − θ), not cos θ = cos(90 − θ). The function changes when you take the complement.
Getting confused with radians. “π/2 ≤ α ≤ π” means Q2, not Q1. Sketch the unit circle quickly if you’re not sure.
Using the wrong sides on the triangle. Sin = opp/hyp. Cos = adj/hyp. Tan = opp/adj. Mix these up and the whole answer is wrong.
Forgetting that two negatives in a fraction give a positive. When both sin and cos are negative (Q3), tan = sin/cos is positive — this is what “tan is positive in Q3” actually means.
Computing tan as cos/sin. No — tan = sin/cos. Sin on top, cos on bottom. (Cos/sin would be cot.)
Once you can confidently jump between sin, cos, and tan and pick the correct sign, the rest of the trig topic becomes a much smoother ride. The next two notes — Linear and Quadratic Trig Equations — both rely on this skill.
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