Exercise 11.3 — Complementary Angles
Trigonometric ratios of complementary angles.
Exercise 11.3 — Trigonometric Ratios of Complementary Angles
Exercise 11.3 continues Chapter 11, Trigonometry, in Class 10 Mathematics (CBSE, Telangana & Andhra Pradesh syllabus) by introducing one of the most powerful shortcuts in the entire chapter: the complementary angle relationships. These six identities let you instantly rewrite any ratio of angle (90° − θ) in terms of θ, which turns many seemingly complicated expressions into simple cancellations.
This exercise has six question types: direct evaluation of expressions using the complementary identities; "show that" proofs where both sides reduce to the same value; solving equations involving 2A and (A − 18°) or two acute angles A and B; a triangle-angle proof connecting half-angles; and finally, expressing an angle-sum expression in terms of angles below 45°.
Deriving the Complementary Angle Identities
Consider a right triangle ABC with ∠B = 90°. The two acute angles, ∠A and ∠C, must add up to 90° (since all three angles of a triangle total 180° and one of them is already 90°). If we call ∠A = θ, then ∠C must equal 90° − θ.
Right triangle ABC with ∠B = 90°, ∠A = θ, and ∠C = 90° − θ (complementary angles).
From this triangle, the six basic ratios for angle θ (at vertex A) are written as usual:
sin θ = BC/AC, cos θ = AB/AC, tan θ = BC/ABcosec θ = AC/BC, sec θ = AC/AB, cot θ = AB/BCNow look at the same triangle from the point of view of angle C, which equals (90° − θ). Because the side opposite to C is AB and the side adjacent to C is BC, the ratios for (90° − θ) come out as the AB and BC versions of the ratios above — but matched to a different name than before:
These six results are the complementary angle identities — the single most useful toolkit in this exercise:
| Identity | In Words |
|---|---|
| sin(90° − θ) = cos θ | sin of the complement equals cos of the angle |
| cos(90° − θ) = sin θ | cos of the complement equals sin of the angle |
| tan(90° − θ) = cot θ | tan of the complement equals cot of the angle |
| cosec(90° − θ) = sec θ | cosec of the complement equals sec of the angle |
| sec(90° − θ) = cosec θ | sec of the complement equals cosec of the angle |
| cot(90° − θ) = tan θ | cot of the complement equals tan of the angle |
Question 1 — Evaluate Using Complementary Angle Identities
This question presents five expressions where the two angles inside always add up to 90°. The strategy every time is identical: rewrite one of the two angles as (90° minus the other), apply the matching identity, and watch the expression collapse to a simple number.
Question 2 — Show That the Following Identities Hold
These two "show that" problems use the same complementary-angle substitution technique as Question 1, but ask you to prove the full equality rather than just compute a final number — meaning you must display both the left-hand side (LHS) and right-hand side (RHS) explicitly to demonstrate they match.
Question 3 — Solving tan 2A = cot(A − 18°)
Here the complementary identity is used in reverse: instead of simplifying a known expression, it is used to convert two different trig functions (tan and cot) into the same function so that the angles themselves can be equated.
Question 4 — Proving A + B = 90° When tan A = cot B
Question 5 — Proving tan((A+B)/2) = cot(C/2) in a Triangle
This question connects the angle sum property of a triangle (the three interior angles always total 180°) with the complementary angle identities, producing a relationship between half-angles that is frequently used in advanced trigonometry problems.
Question 6 — Expressing sin 75° + cos 65° Using Angles Below 45°
This question asks for an expression to be rewritten entirely using angles between 0° and 45°. Since both 75° and 65° are above 45°, each one needs to be converted into its complementary form so that the angle inside each function drops below the 45° threshold.
Common Mistakes to Avoid
- Swapping the wrong pair: Remember the three fixed pairs — sin↔cos, tan↔cot, sec↔cosec. Mixing up which function pairs with which (e.g. confusing tan↔sec) leads to an incorrect identity.
- Forgetting to check the angle sum first: Before applying any complementary identity, always confirm the two angles in the question genuinely add up to 90° — the technique only works for true complementary pairs.
- Sign errors when rearranging equations: In Question 3, carefully expand 90° − 2A = A − 18° step by step; rushing this rearrangement is the most common source of wrong answers.
- Stopping at the substitution instead of finishing the proof: In "show that" questions (Question 2), always explicitly write out the final LHS = RHS line — a half-finished substitution is not considered a complete proof in board exams.
- Confusing reciprocal identities with complementary identities: tanθ = 1/cotθ is a reciprocal identity (same angle), while tan(90°−θ) = cotθ is a complementary identity (different angle) — both are used together in parts (iv) and (v) of Question 1, so keep them distinct.
- Not simplifying half-angle expressions fully: In Question 5, make sure to divide both sides of A + B = 180° − C by 2 correctly before applying the tan ratio.
Quick Reference — All Answers at a Glance
| Question | Topic | Key Result |
|---|---|---|
| Q1(i) | tan 36° ÷ cot 54° | 1 |
| Q1(ii) | cos 12° − sin 78° | 0 |
| Q1(iii) | cosec 31° − sec 59° | 0 |
| Q1(iv) | sin 15° · sec 75° | 1 |
| Q1(v) | tan 26° · tan 64° | 1 |
| Q2(i) | tan48°tan16°tan42°tan74° | = 1, proved |
| Q2(ii) | cos36cos54 − sin36sin54 | = 0, proved |
| Q3 | tan2A = cot(A−18°) | A = 36° |
| Q4 | tanA = cotB | A+B=90°, proved |
| Q5 | Triangle half-angle identity | tan((A+B)/2)=cot(C/2), proved |
| Q6 | sin75° + cos65° | = cos15° + sin25° |
What This Exercise Prepares You For
Exercise 11.3 builds the algebraic flexibility needed for the rest of the Trigonometry chapter. The complementary angle identities practised here are used constantly when proving the fundamental trigonometric identities and when simplifying mixed-angle expressions in later exercises. The standard-angle values from Exercise 11.2 combine directly with these complementary rules — for instance, sin 30° = cos 60° is simply the complementary identity in action at a specific angle.
The angle-sum reasoning used in Question 5 also reinforces the triangle angle-sum property from Triangles, while the equation-solving technique in Questions 3 and 4 is a direct preview of the algebraic manipulation skills needed in Trigonometric Identities, the next major topic in this chapter.