Trigonometry Functions and Identities
~14 min read
- What: Trigonometric ratios in a right triangle (sin, cos, tan, cosec, sec, cot), values at standard angles (0°, 30°, 45°, 60°, 90°), the three Pythagorean identities, complementary-angle relations, and simplification problems.
- Why it matters: Among the most-tested CDS Maths chapters — 6–10 questions per paper across trigonometry and its application (Height and Distance).
- Key fact: Three identities anchor everything: \(\sin^2\theta + \cos^2\theta = 1\), \(1 + \tan^2\theta = \sec^2\theta\), \(1 + \cot^2\theta = \csc^2\theta\). Memorise these and 80% of CDS trigonometry simplification collapses.
Trigonometry is one of the highest-yield CDS chapters. The formulas are tight, the values at standard angles are memorisable in an evening, and the simplification archetypes recur paper after paper. Get the three Pythagorean identities and the complementary-angle table cold, and you collect 6–10 marks per paper between this chapter and Height and Distance.
This page is built from CDS Previous Year Questions across 2002–2023 plus NCERT Class 10 Introduction to Trigonometry. Pair with Height and Distance (the application chapter).
What This Topic Covers
CDS scope: (1) the six ratios — sin, cos, tan, cosec, sec, cot; (2) standard angle values — 0°, 30°, 45°, 60°, 90°; (3) the three Pythagorean identities; (4) complementary angles — \(\sin(90° - \theta) = \cos\theta\), etc.; (5) simplification — expressions involving multiple ratios; (6) trigonometric equations — find \(\theta\) such that an equation holds; and (7) algebraic manipulation — given \(\sin\theta + \cos\theta = k\), find \(\sin\theta \cos\theta\), etc.
Why This Topic Matters
- Single largest weightage in the geometry/trigonometry block of CDS Maths.
- Direct application in Height and Distance, which alone carries 2-3 marks per paper.
- Algebraic identity manipulation (given \(\sin\theta + \cos\theta\), find \(\sin^3\theta + \cos^3\theta\)) is a high-yield 30-second move.
Exam Pattern & Weightage
| Year / Paper | No. | Subtopics Tested |
|---|---|---|
| 2007-II | 4 | Standard angles, simplification |
| 2008-I/II | 4 | Complementary angles, identities |
| 2009-II | 4 | Algebraic manipulation, identities |
| 2010-I/II | 5 | Sin-cos-tan values, identity |
| 2011-I/II | 5 | Complementary, simplification |
| 2012-I/II | 5 | Pythagorean, algebraic |
| 2013-I/II | 6 | Mixed identities, complementary |
| 2014-I/II | 5 | Find \(\theta\), simplification |
| 2015-I/II | 5 | Sin/cos given, find others |
| 2016-I/II | 5 | Algebraic, complementary |
| 2017-I/II | 6 | Mixed |
| 2018-I/II | 5 | Identities, simplification |
| 2019-II | 4 | Standard values, find \(\theta\) |
| 2020-I/II | 5 | Mixed |
| 2021-I/II | 5 | Algebraic manipulation |
| 2022-I / 2023-I | 5 | Mixed |
For \(\sin\theta + \cos\theta = k\), square both sides: \(\sin^2\theta + \cos^2\theta + 2\sin\theta\cos\theta = k^2 \implies 1 + 2\sin\theta\cos\theta = k^2 \implies \sin\theta\cos\theta = (k^2 - 1)/2\). This identity solves dozens of CDS algebraic-trig problems.
Core Concepts
The Six Ratios in a Right Triangle
In a right triangle with angle \(\theta\), opposite \(O\), adjacent \(A\), hypotenuse \(H\):
Standard Angle Values
| θ | sin | cos | tan | cosec | sec | cot |
|---|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | ∞ | 1 | ∞ |
| 30° | 1/2 | √3/2 | 1/√3 | 2 | 2/√3 | √3 |
| 45° | 1/√2 | 1/√2 | 1 | √2 | √2 | 1 |
| 60° | √3/2 | 1/2 | √3 | 2/√3 | 2 | 1/√3 |
| 90° | 1 | 0 | ∞ | 1 | ∞ | 0 |
The Three Pythagorean Identities
Complementary-Angle Relations
Reciprocal and Quotient Relations
\(\sin^2\theta\) means \((\sin\theta)^2\), not \(\sin(\theta^2)\). The square is applied to the value, not the angle. CDS uses the notation freely — read carefully.
Algebraic Manipulation Identities
Worked Examples
Example 1 — Standard Angle (2010-I)
Q: Find the value of \(\sin 30° + \cos 60°\).
- \(\sin 30° = 1/2\) and \(\cos 60° = 1/2\). Sum = 1.
Example 2 — Pythagorean Simplification (2011-I)
Q: Simplify \(\sin^2 60° + \cos^2 60°\).
- By Pythagorean identity, for any angle: \(\sin^2\theta + \cos^2\theta = 1\). So answer = 1.
Example 3 — Complementary Angle (2008-I)
Q: Simplify \(\sin 35° \cdot \cos 55° + \cos 35° \cdot \sin 55°\).
- Note \(35° + 55° = 90°\). Use \(\cos 55° = \sin 35°\) and \(\sin 55° = \cos 35°\).
- Expression = \(\sin^2 35° + \cos^2 35° = 1\).
Example 4 — Find \(\theta\) (2014-I)
Q: If \(\tan\theta = \sqrt{3}\), find \(\theta\).
- From standard table: \(\tan 60° = \sqrt{3}\). So \(\theta = 60°\).
Example 5 — Algebraic from \(\sin + \cos\) (2016-II)
Q: If \(\sin\theta + \cos\theta = \sqrt{2}\), find \(\sin\theta \cos\theta\).
- Square: \(\sin^2\theta + \cos^2\theta + 2\sin\theta\cos\theta = 2 \implies 1 + 2\sin\theta\cos\theta = 2 \implies \sin\theta\cos\theta = 1/2\).
Example 6 — Find Other Ratios (2017-II)
Q: If \(\sin\theta = 3/5\), find \(\cos\theta\) and \(\tan\theta\).
- \(\cos\theta = \sqrt{1 - 9/25} = \sqrt{16/25} = 4/5\).
- \(\tan\theta = \sin/\cos = 3/4\).
Example 7 — Simplification (2018-I)
Q: Simplify \(\tan\theta \cot\theta + \sec\theta \csc\theta\) at \(\theta = 45°\).
- \(\tan 45° \cot 45° = 1 \cdot 1 = 1\). \(\sec 45° \csc 45° = \sqrt{2} \cdot \sqrt{2} = 2\).
- Sum = 3. Alternative: \(\tan\theta \cot\theta = 1\) always (reciprocals).
How CDS Tests This Topic
Six recurring archetypes: (1) plug standard angle values into a given expression, (2) apply Pythagorean identity for simplification, (3) complementary angle simplification (typically with \(\theta\) and \(90° - \theta\)), (4) "find \(\theta\)" from a given ratio value, (5) given one ratio, find others (Pythagorean triangle), (6) algebraic — given \(\sin + \cos\), find \(\sin \cos\), \(\sin^3 + \cos^3\), etc.
Exam Shortcuts (Pro-Tips)
Shortcut 1 — Standard Angle Table Cold
Memorise the table for 0°, 30°, 45°, 60°, 90°. These five rows answer 50% of CDS trig questions immediately.
Shortcut 2 — \(\sin\theta + \cos\theta = k\) Square Trick
Shortcut 3 — Complementary Reflex
When angles in a problem sum to 90°, look for the complementary identity. \(\sin a \cos b + \cos a \sin b\) where \(a + b = 90°\) reduces to 1.
Shortcut 4 — Pythagorean Triangle for Other Ratios
Given \(\sin\theta = a/c\), draw a right triangle with opposite \(a\), hypotenuse \(c\); compute adjacent \(b = \sqrt{c^2 - a^2}\). Then read off all six ratios.
Shortcut 5 — Identity Triangle Recognition
Recognise 3-4-5, 5-12-13, 8-15-17 triangles instantly. They give clean sin, cos, tan values.
Common Question Patterns
Pattern 1 — Standard Angle Evaluation
Plug 0°, 30°, 45°, 60°, 90° into a given expression. Reach a clean answer.
Pattern 2 — Pythagorean Simplification
An expression collapses to 1, 0, or a known constant via \(\sin^2 + \cos^2 = 1\) or its cousins.
Pattern 3 — Complementary Angles
Angles summing to 90°. Use \(\sin(90° - \theta) = \cos\theta\), etc.
Pattern 4 — Find Other Ratios from One
Given \(\sin\theta = a/c\). Build Pythagorean triangle. Read off others.
Pattern 5 — Algebraic Manipulation
Given \(\sin + \cos = k\) (or sin·cos, or any combination), find sin³+cos³ or similar via squaring and identities.
Preparation Strategy
Week 1. Memorise the standard angle table cold. Drill 20 standard-evaluation problems. Practice the three Pythagorean identities.
Week 2. Complementary angles, find-other-ratios, and algebraic manipulation. Practice the "square the sum" trick on 10 problems.
Week 3. Integrate with Height and Distance. Apply trig to real-world angles of elevation and depression.
Use CDS mock tests for timed practice. Most CDS slip-ups in trig are sign or value errors at standard angles — drill until reflex.
Drill Trigonometry at Speed
CDS mocks with standard angles, Pythagorean simplification, and algebraic manipulation. Six archetypes — three identities — reflex.
Start Free Mock TestFrequently Asked Questions
What are the three Pythagorean identities?
\(\sin^2\theta + \cos^2\theta = 1\), \(1 + \tan^2\theta = \sec^2\theta\), \(1 + \cot^2\theta = \csc^2\theta\). All three derive from the basic right-triangle Pythagoras theorem.
What are the values of sin, cos, tan at standard angles?
\(\sin\) goes 0, 1/2, 1/√2, √3/2, 1 from 0° to 90°. \(\cos\) goes the reverse. \(\tan\) goes 0, 1/√3, 1, √3, ∞. Memorise the table — it's the single most important asset for CDS trig.
What is the complementary-angle relationship?
For \(\theta\) and \(90° - \theta\): \(\sin\) and \(\cos\) swap; \(\tan\) and \(\cot\) swap; \(\sec\) and \(\csc\) swap. Example: \(\sin 60° = \cos 30°\).
If I know \(\sin + \cos\), how do I find \(\sin \cdot \cos\)?
Square: \((\sin\theta + \cos\theta)^2 = \sin^2\theta + \cos^2\theta + 2\sin\theta\cos\theta = 1 + 2\sin\theta\cos\theta\). So \(\sin\theta\cos\theta = ((\sin + \cos)^2 - 1)/2\). Example: if \(\sin + \cos = \sqrt{2}\), then \(\sin\cos = 1/2\).
If \(\sin\theta = 3/5\), what is \(\cos\theta\)?
Use \(\cos\theta = \sqrt{1 - \sin^2\theta} = \sqrt{1 - 9/25} = \sqrt{16/25} = 4/5\). For an acute angle, cosine is positive. The 3-4-5 triangle gives \(\sin = 3/5\), \(\cos = 4/5\), \(\tan = 3/4\).
What does \(\sin^2\theta\) mean?
\(\sin^2\theta\) means \((\sin\theta)^2\) — the square of the sine of \(\theta\). It does not mean \(\sin(\theta^2)\). The notation is shorthand; CDS uses it freely.
Which CDS Maths topics use Trigonometry directly?
Height and Distance — pure application of trig ratios on angles of elevation/depression. Triangles — many properties involve trig ratios. Circles — arc-and-chord problems use trig.