Class 10 Maths Chapter 8: Introduction To Trigonometry Notes

Key Formulas and Concepts to Remember

• Trigonometric Ratios (SOH CAH TOA): sin A = Opposite / Hypotenuse cos A = Adjacent / Hypotenuse * tan A = Opposite / Adjacent
• Reciprocal Ratios: cosec A = 1/sin A = Hypotenuse / Opposite sec A = 1/cos A = Hypotenuse / Adjacent * cot A = 1/tan A = Adjacent / Opposite
• Quotient Identities: tan A = sin A / cos A cot A = cos A / sin A
• Pythagorean Identities: sin² A + cos² A = 1 1 + tan² A = sec² A * 1 + cot² A = cosec² A
• Trigonometric Ratios of Complementary Angles: sin (90° - A) = cos A cos (90° - A) = sin A tan (90° - A) = cot A cot (90° - A) = tan A sec (90° - A) = cosec A cosec (90° - A) = sec A
• The value of sin A increases as A increases from 0° to 90°. The value of cos A decreases as A increases from 0° to 90°.
• Trigonometric ratios are unitless as they are ratios of lengths.
• An angle can be expressed in degrees or radians. For Class 10, primarily degrees are used.

Essential Trigonometric Terms

Trigonometry

The branch of mathematics that studies relationships between side lengths and angles of triangles.

Trigonometric Ratios

Ratios of the sides of a right-angled triangle with respect to its acute angles. There are six primary ratios: sine, cosine, tangent, cosecant, secant, and cotangent.

Hypotenuse

The longest side in a right-angled triangle, always opposite the right angle.

Opposite Side

The side of a right-angled triangle directly across from a given acute angle.

Adjacent Side

The side of a right-angled triangle next to a given acute angle, which is not the hypotenuse.

Trigonometric Identities

Equations involving trigonometric ratios that are true for all valid values of the angles involved. Examples include Pythagorean identities.

Complementary Angles

Two angles whose sum is 90 degrees. Trigonometric ratios of complementary angles are inter-related.

Understanding Trigonometric Ratios in a Right Triangle

Trigonometry begins with the study of right-angled triangles. In any right-angled triangle, one angle measures 90 degrees. The other two angles are acute (less than 90 degrees). For a specific acute angle (let's call it 'A'), we define three key sides relative to it:

1. Hypotenuse: This is always the longest side, and it's always opposite the right angle. Its length remains constant regardless of which acute angle you're focusing on.
2. Opposite Side: This is the side directly across from the angle 'A' you are considering.
3. Adjacent Side: This is the side next to angle 'A', which is not the hypotenuse.

Once these sides are identified, we can define the six trigonometric ratios. A popular mnemonic to remember the primary three is SOH CAH TOA:

• SOH: Sine = Opposite / Hypotenuse
• CAH: Cosine = Adjacent / Hypotenuse
• TOA: Tangent = Opposite / Adjacent

The other three ratios (cosecant, secant, cotangent) are simply the reciprocals of sine, cosine, and tangent, respectively. For example, if sin A = Opposite/Hypotenuse, then cosec A = Hypotenuse/Opposite. These ratios are fundamental because they establish a precise relationship between the angles and the proportional lengths of the sides in any right-angled triangle, allowing us to find unknown angles or sides if some information is given. Remember, these ratios are unitless as they are ratios of lengths.

Trigonometric Ratios of Specific Angles (0°, 30°, 45°, 60°, 90°)

Worked Examples

• {"title":"Example 1: Finding Ratios","bodyMarkdown":"Q: If sin A = 3/5, find cos A and tan A.\n\nA: We know that sin² A + cos² A = 1.\n(3/5)² + cos² A = 1\n9/25 + cos² A = 1\ncos² A = 1 - 9/25 = 16/25\ncos A = √(16/25) = 4/5 (assuming A is acute).\nNow, tan A = sin A / cos A = (3/5) / (4/5) = 3/4."}
• {"title":"Example 2: Using Identities","bodyMarkdown":"Q: Evaluate: (1 - tan² 45°) / (1 + tan² 45°)\n\nA: We know that tan 45° = 1.\nSubstitute the value: (1 - (1)²) / (1 + (1)²) = (1 - 1) / (1 + 1) = 0 / 2 = 0."}

Exam Tips for Trigonometry

1. Memorize Identities and Values: The Pythagorean identities (sin²A + cos²A = 1, etc.) and the table of specific angle values (0°, 30°, 45°, 60°, 90°) are your best friends. Practice writing them down until they're second nature.

2. Draw Diagrams: For problems involving right-angled triangles where ratios are given, always draw a clear diagram. Label the opposite, adjacent, and hypotenuse correctly with respect to the angle in question. This prevents silly errors.

3. Algebraic Manipulation: Many trigonometry problems boil down to algebraic simplification after applying identities. Be careful with squaring terms, taking square roots, and combining fractions. Show all steps clearly to earn partial marks.

4. Check for Complementary Angles: Look for expressions like sin(90° - A) or cos(90° - B). Using complementary angle identities often simplifies complex expressions significantly.

5. Practice Proofs: For identity proofs, always start from one side (LHS or RHS) and transform it into the other. Avoid manipulating both sides simultaneously unless you're trying to prove LHS - RHS = 0. Use common denominators for fractions.

Quick Revision Check

• Q: What is the value of (sin 60° cos 30°) + (sin 30° cos 60°)? A: (√3/2 √3/2) + (1/2 1/2) = (3/4) + (1/4) = 4/4 = 1.
• Q: If tan A = cot B, what is the relationship between angles A and B (assuming they are acute)? A: If tan A = cot B, then tan A = tan (90° - B). Therefore, A = 90° - B, which means A + B = 90°. A and B are complementary angles.
• Q: State one Pythagorean trigonometric identity. A: sin² θ + cos² θ = 1 (or 1 + tan² θ = sec² θ, or 1 + cot² θ = cosec² θ).
• Q: Can the value of sin A be greater than 1? Justify your answer. A: No, the value of sin A cannot be greater than 1. In a right-angled triangle, sin A = Opposite / Hypotenuse. Since the hypotenuse is always the longest side, the opposite side can never be greater than the hypotenuse, so the ratio will always be less than or equal to 1.

Frequently Asked Questions