Trigonometry is the branch of mathematics that studies relationships between the sides and angles of triangles. For MP TET Varg-2, this topic focuses on right-angled triangles and the six trigonometric ratios that relate an acute angle to the ratios of two sides of the triangle.
This topic forms a bridge between geometry and algebra, appearing consistently in the Mathematics and Science paper. Questions typically test your ability to recall ratio definitions, apply standard identities, and solve for unknown sides or angles. Mastery requires memorising the ratio definitions, understanding the relationships between ratios, and practising identity-based simplifications.
Students must be comfortable with the Pythagorean theorem (Hypotenuse² = Base² + Perpendicular²) before tackling trigonometry, as it underpins many derivations and problem solutions.
Key Concepts
**Right-angled triangle orientation**: For a given acute angle θ in a right triangle, identify three sides — the side opposite to θ (Perpendicular), the side adjacent to θ (Base), and the longest side opposite the right angle (Hypotenuse).
**Six trigonometric ratios**: These are defined as ratios of specific pairs of sides with respect to angle θ. The three primary ratios are sine, cosine, and tangent; the three secondary ratios are their reciprocals.
**Reciprocal relationships**: Cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent. This means sin θ × cosec θ = 1, and similarly for the other pairs.
**Quotient relationships**: tan θ = sin θ / cos θ and cot θ = cos θ / sin θ. These help convert between ratios during simplification.
**Pythagorean identities**: Three fundamental identities derived from the Pythagorean theorem connect the squares of trigonometric ratios.
**Standard angle values**: The ratios for 0°, 30°, 45°, 60°, and 90° must be memorised as they appear directly in exam questions.
**Complementary angle relations**: The trigonometric ratio of an angle equals a different ratio of its complement (90° − θ). For example, sin θ = cos(90° − θ).
Formulas / Key Facts
**Trigonometric Ratio Definitions** (for acute angle θ in a right triangle):
| Ratio | Definition | |-------|------------| | sin θ | Perpendicular / Hypotenuse | | cos θ | Base / Hypotenuse | | tan θ | Perpendicular / Base | | cosec θ | Hypotenuse / Perpendicular = 1 / sin θ | | sec θ | Hypotenuse / Base = 1 / cos θ | | cot θ | Base / Perpendicular = 1 / tan θ |
**Memory trick**: "Some People Have Curly Brown Hair"
**Memory pattern for sin values**: √0/2, √1/2, √2/2, √3/2, √4/2 (i.e., 0, 1/2, 1/√2, √3/2, 1) **cos values**: Reverse order of sin values
Worked Examples
**Example 1**: In a right triangle ABC with right angle at B, if AC = 13 cm and BC = 5 cm, find sin A, cos A, and tan A.
*Solution*:
AC is the hypotenuse (opposite to right angle B) = 13 cm
BC is the side adjacent to angle A (Base) = 5 cm
Using Pythagorean theorem: AB² + BC² = AC²
AB² + 25 = 169
AB² = 144, so AB = 12 cm (Perpendicular to angle A)
Therefore:
sin A = Perpendicular/Hypotenuse = 12/13
cos A = Base/Hypotenuse = 5/13
tan A = Perpendicular/Base = 12/5
**Example 2**: Prove that (sin²30° + cos²30°) = 1
*Solution*:
sin 30° = 1/2, so sin²30° = 1/4
cos 30° = √3/2, so cos²30° = 3/4
sin²30° + cos²30° = 1/4 + 3/4 = 4/4 = 1
This verifies the Pythagorean identity.
**Example 3**: Simplify: tan 45° + cot 45° − sin 90°
*Solution*:
tan 45° = 1
cot 45° = 1
sin 90° = 1
Expression = 1 + 1 − 1 = 1
**Example 4**: If sin θ = 3/5, find cos θ and tan θ (where θ is acute).
*Solution*: Using sin²θ + cos²θ = 1:
(3/5)² + cos²θ = 1
9/25 + cos²θ = 1
cos²θ = 16/25
cos θ = 4/5 (positive since θ is acute)
tan θ = sin θ / cos θ = (3/5) / (4/5) = 3/4
Common Mistakes
**Confusing sides relative to the angle**: Students often mix up which side is perpendicular and which is base. Fix: Always identify sides with respect to the specific angle in question, not the triangle in general. The perpendicular is always opposite to the angle you are considering.
**Misremembering standard values**: Swapping sin and cos values for 30° and 60° is extremely common. Fix: Remember that sin increases from 0° to 90° (0 to 1), while cos decreases (1 to 0). At 30°, sin is smaller (1/2) and cos is larger (√3/2).
**Forgetting tan 90° is undefined**: Students sometimes write tan 90° = 1 or 0. Fix: Since cos 90° = 0 and tan = sin/cos, division by zero makes tan 90° undefined.
**Wrong application of Pythagorean identity**: Writing sin²θ + cos²θ = 2 or using sin θ + cos θ = 1. Fix: The identity involves squares of the ratios, not the ratios themselves. sin θ + cos θ ≠ 1 in general.
**Ignoring the complementary angle relationship**: When seeing sin 60°, not recognising it equals cos 30°. Fix: Practise converting between complementary forms; this simplifies many exam problems.
Quick Reference
sin θ = P/H, cos θ = B/H, tan θ = P/B (SOH-CAH-TOA)
sin²θ + cos²θ = 1 — the most important identity
sin 30° = cos 60° = 1/2; sin 60° = cos 30° = √3/2; sin 45° = cos 45° = 1/√2
tan θ = sin θ / cos θ; cot θ = 1/tan θ
Complementary rule: sin θ = cos(90° − θ) and vice versa
At 45°, all primary ratios involving equal sides yield 1 for tan and 1/√2 for sin and cos
You read the notes — now try one
In a right-angled triangle ABC, right-angled at B, if AB = 3 cm and BC = 4 cm, what is the value of sin C?
Tap an option to check your answer.
👥 Study this together
Invite your prep group — read the same notes, then discuss doubts in this topic's shared room.