Introduction to trigonometry
The chapter 'Introduction to trigonometry' carrying the weightage of 6-7 marks in the board examination and forms the base for higher studies. It covers the concepts like trigonometric ratios, standard angles and trigonometric indentities.
Most possible variation of question types that we can expect in board exam as per previous year question paper are discussed below.
| Total Marks 6 to 7 | |
| Variation 1 | Variation 2 |
Total marks = 6
|
Total marks = 7
|
To prepare well for the board examination, it is necessary to understand the following concepts clearly.
- Trigonometric ratios - Understand the definitions of sine, cosine, tangent and their reciprocals.
- Standard trigonometric angles - Know how to use the values of trigonometric standard angles.
- Trigonometric identites - Know about the basic trigonometric identites and their rearrangements.
|
Important concept
(Learning Outcomes)
|
Expected Question Type | Concept dealt with |
| Trigonometric ratios | Sec A, Sec B | Find the ratios |
| Trigonometric standard values | Sec A, Sec B, Sec C | |
| Trigonometric identities | Sec B, Sec D |
Let us recall the concepts in introduction to trigonometry:
1. Trigonometric identities:
\(Sin^2\theta+cos^2\theta = 1\)
\(1+ tan^2\theta = sec^2\theta\)
\(1+ cot^2\theta = cosec^2\theta\)
\(Sin^2\theta+cos^2\theta = 1\)
\(1+ tan^2\theta = sec^2\theta\)
\(1+ cot^2\theta = cosec^2\theta\)
2. Trigonometric ratios:
| \(sin \theta = \frac{opposite\ side}{hypotenus}\) | \(cosec \theta = \frac{hypotenus}{opposite\ side}\) |
| \(cos \theta = \frac{Adjacent\ side}{hypotenus}\) | \(sec \theta = \frac{Adjacent\ side}{hypotenus}\) |
| \(tan \theta = \frac{opposite\ side}{Adjacent\ side}\) | \(cot \theta = \frac {Adjacent\ side}{opposite\ side}\) |
3. Reciprocal of trigonometry:
| \(sin \theta = \frac{1}{cosec\theta}\) | \(cosec\theta = \frac{1}{sin\theta}\) |
| \(cos\theta = \frac{1}{sec\theta}\) | \(sec\theta = \frac{1}{cos\theta}\) |
| \(tan\theta = \frac{1}{cot\theta}\) | \(cot\theta = \frac {1}{tan\theta}\) |
| \(tan\theta = \frac{sin\theta}{cos\theta}\) | \(cot\theta = \frac{cos\theta}{sin\theta}\) |
| \(tan\theta = \frac{sec\theta}{cosec\theta}\) | \(cot\theta = \frac{cosec\theta}{sec\theta}\) |
4. Velues of trigonometric ratios:
|
Trignometric ratio
|
0° | 30° | 45° | 60° | 90° |
| \(sin \theta\) | \(0\) | \(\frac{1}{2}\) | \(\frac{1}{\sqrt{2}}\) | \(\frac{\sqrt{3}}{2}\) | \(1\) |
| \(cos \theta\) | \(1\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{\sqrt{2}}\) | \(\frac{1}{2}\) | \(0\) |
| \(tan \theta\) | \(0\) | \(\frac{1}{\sqrt{3}}\) | \(1\) | \(\sqrt{3}\) | not defined |
| \(cosec \theta\) | not defined | \(2\) | \(\sqrt{2}\) | \(\frac{2}{\sqrt{3}}\) | \(1\) |
| \(sec \theta\) | \(1\) | \(\frac{2}{\sqrt{3}}\) | \(\sqrt{2}\) | \(2\) | not defined |
| \(cot \theta\) | not defined | \(\sqrt{3}\) | \(1\) | \(\frac{1}{\sqrt{3}}\) | \(0\) |
Common mistakes to avoid:
1. Confusing perpendicular and base: Identifying perpendicular and base fixed to the visual diagram rather than relative to the given angle \(\theta\).
2. Incorrect notation & powers: Writing \((sin\ \theta)^2\) as \(sin\ \theta^2\) (squaring the angle instead of the ratio)
3. Forgetting to square fractions when replacing standard values into squared terms.
4. Incorrectly assuming the square roots on sums [e.g., \(\sqrt{sin^2\ \theta + cos^2\ \theta} = sin\ \theta + cos\ \theta\). Always apply identity laws first.
Exam tips & tricks:
1. The \(sin\ \theta \) value generation trick: Quickly write down values for \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), \(90^\circ\) using \(\sqrt{\frac{0}{4}}\), \(\sqrt{\frac{1}{4}}\), \(\sqrt{\frac{2}{4}}\), \(\sqrt{\frac{3}{4}}\), \(\sqrt{\frac{4}{4}}\), giving \(0\), \(\frac{1}{2}\), \(\frac{1}{\sqrt{2}}\), \(\frac{\sqrt{3}}{2}\), \(1\). Reverse the order for \(cos\ \theta\).
2. Converting everything to \(sin\ \theta\) & \(cos\ \theta\): When soling identity proofs, express all \(tan\), \(cot\), \(sec\), \(cosec\) terms strictly in terms of \(sin\ \theta\) and \(cos\ \theta\) as your first step.
3. To quickly verify identity questions or options in MCQs, substitute simple test angles like \(\theta = 30^\circ\) or \(45^\circ\) into both sides of the expression.
4. Clean Step-Wise Proof Layout: Keep LHS and RHS separate during identity proofs. Start explicitly with \(\text{LHS} = \dots\), work through algebraic steps line-by-line, and conclude with \(= \text{RHS}\) followed by "Hence Proved."