Trigonometric Ratios

Consider a right-angled triangle ABC with the right angle at C. Let angle A = theta (an acute angle). Then:

The side opposite to angle theta is called the opposite side (or perpendicular, P) = BC.

The side adjacent to angle theta is called the adjacent side (or base, B) = AC.

The longest side (opposite the right angle) is the hypotenuse (H) = AB.

The Six Trigonometric Ratios:

Memory Aid (Mnemonic): SOH-CAH-TOA

Sin = Opposite / Hypotenuse
Cos = Adjacent / Hypotenuse
Tan = Opposite / Adjacent

Key Points:

• Trigonometric ratios are defined only for acute angles (0 < theta < 90 degrees) in Class 10.
• The ratios depend on the angle, NOT on the size of the triangle. Two right triangles with the same acute angle theta will have the same trigonometric ratios (by similar triangles).
• The hypotenuse is always the longest side, so sin theta and cos theta are always between 0 and 1 (for acute angles).
• tan theta can take any positive value for acute angles.

Derivation of Trigonometric Identities:

Identity 1: sin^2(theta) + cos^2(theta) = 1

Consider a right triangle with sides P (opposite), B (adjacent), H (hypotenuse), and acute angle theta.

By the Pythagoras Theorem: P^2 + B^2 = H^2

Dividing both sides by H^2: P^2/H^2 + B^2/H^2 = 1

(P/H)^2 + (B/H)^2 = 1

sin^2(theta) + cos^2(theta) = 1. QED.

Identity 2: 1 + tan^2(theta) = sec^2(theta)

Starting from P^2 + B^2 = H^2, divide by B^2:

P^2/B^2 + 1 = H^2/B^2

tan^2(theta) + 1 = sec^2(theta). QED.

Identity 3: 1 + cot^2(theta) = cosec^2(theta)

Starting from P^2 + B^2 = H^2, divide by P^2:

1 + B^2/P^2 = H^2/P^2

1 + cot^2(theta) = cosec^2(theta). QED.

Why Trigonometric Ratios Depend Only on the Angle:

Consider two right triangles with the same acute angle theta but different sizes. By the AA similarity criterion (both have a right angle and the common angle theta), the triangles are similar. In similar triangles, corresponding sides are proportional, so the ratio P/H (sin theta) is the same for both triangles. This is why trigonometric ratios are properties of the angle, not of any particular triangle.

Derivation of Complementary Angle Relations:

In right triangle ABC with right angle at C: angle A + angle B = 90 degrees. So angle B = 90 - angle A.

sin A = BC/AB = opposite of A / hypotenuse = P/H.

cos B = cos(90 - A) = BC/AB = P/H (because BC is adjacent to B).

Therefore sin A = cos(90 - A), and cos A = sin(90 - A).

Problems on trigonometric ratios in Class 10 fall into several categories:

Type 1: Finding All Ratios from One Given Ratio

Given one trigonometric ratio (e.g., sin theta = 3/5), find all other five ratios. Use the Pythagoras Theorem to find the missing side, then compute all ratios.

Type 2: Proving Trigonometric Identities

Prove algebraic equalities involving trigonometric ratios. Use the three Pythagorean identities, reciprocal relations, and quotient relations.

Type 3: Evaluating Trigonometric Expressions

Calculate the value of expressions involving trigonometric ratios of specific angles.

Type 4: Solving for Unknown Angles

Given a trigonometric equation like sin theta = 1/2, find the angle theta (within the acute angle range).

Type 5: Complementary Angle Problems

Use the relations sin(90 - theta) = cos theta, etc., to simplify or evaluate expressions.

Type 6: Word Problems Involving Right Triangles

Apply trigonometric ratios to find unknown sides or angles in real-world right triangle situations (heights, distances, angles of elevation/depression).

Method 1: Finding All Ratios from One Ratio

Given sin theta = 3/5. Then P = 3k, H = 5k for some positive k. By Pythagoras: B = sqrt(H^2 - P^2) = sqrt(25k^2 - 9k^2) = 4k.

cos theta = B/H = 4/5, tan theta = P/B = 3/4, cosec theta = 5/3, sec theta = 5/4, cot theta = 4/3.

Method 2: Proving Identities (LHS = RHS)

Start with one side (usually the more complex side). Simplify using: sin^2 + cos^2 = 1, 1 + tan^2 = sec^2, 1 + cot^2 = cosec^2, and the reciprocal/quotient relations. Try to reach the other side.

Method 3: Using Complementary Angles

Replace sin(90-A) with cos A, tan(90-A) with cot A, etc. This often simplifies expressions dramatically.

Example: sin 65 / cos 25 = sin 65 / cos(90-65) = sin 65 / sin 65 = 1.

Method 4: Finding Angles from Ratio Values

Use the known values: sin 30 = 1/2, cos 60 = 1/2, tan 45 = 1, sin 60 = sqrt(3)/2, etc. Match the given ratio to identify the angle.

Tips:

• Always identify which angle is theta before labelling sides as opposite, adjacent, and hypotenuse.
• The hypotenuse is always opposite the right angle and is always the longest side.
• sin and cos are always between 0 and 1 for acute angles. If you compute a value greater than 1, you have made an error.
• tan theta is sin theta / cos theta. Use this when converting between ratios.
• When proving identities, never move terms from one side to the other. Work on one side only.

Trigonometric ratios have an enormous range of applications in science, engineering, and everyday life.

Heights and Distances: Trigonometry is used to calculate the height of a building, mountain, or tower by measuring the angle of elevation from a known distance. Similarly, the depth of a valley or the distance of a ship from a lighthouse is found using the angle of depression. These problems are covered in detail in Class 10 Chapter 9 (Applications of Trigonometry).

Architecture and Construction: Engineers use trigonometric ratios to calculate the slope of a roof, the angle of a ramp, the height of a building from its shadow, and the load distribution on inclined surfaces. The pitch of a roof is expressed as a ratio (rise/run), which is the tangent of the angle of inclination.

Navigation: Ships and aircraft use trigonometric ratios to calculate bearings, headings, and distances. The course of a ship is expressed as an angle, and trigonometric ratios convert this angle into north-south and east-west components of motion.

Physics: Trigonometric ratios are fundamental in resolving forces into components (horizontal and vertical), analysing wave motion (amplitude, frequency, phase), calculating the trajectory of projectiles, and understanding oscillations (pendulums, springs). Snell's law in optics uses the sine ratio to describe the bending of light at surfaces.

Astronomy: The distances to celestial objects are calculated using trigonometric parallax. The altitude and azimuth of stars are described using trigonometric ratios, enabling astronomers to track celestial objects across the sky.

Music and Sound: Sound waves are described mathematically using sine and cosine functions. The pitch, volume, and timbre of musical notes are analysed using trigonometric decomposition (Fourier analysis), which represents any complex waveform as a sum of sine and cosine waves.

Computer Graphics and Animation: Rotation of objects in 2D and 3D is performed using sine and cosine values. Every rotated pixel on a computer screen has its new position calculated using trigonometric ratios. Game engines use trigonometry extensively for character movement, camera angles, and collision detection.