Introduction to Trigonometry

Trigonometry is the study of relationships between the sides and angles of a triangle. It is applied in various fields like astronomy, engineering, and physics to calculate distances and heights. For example, it can determine the height of a building without direct measurement using right-angled triangles.

The six trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot). They are defined as ratios of sides of a right-angled triangle relative to one of its acute angles. For example, sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, and tan A = opposite/adjacent.

Sin A is the ratio of the length of the side opposite to angle A to the hypotenuse, while cosec A is the reciprocal of sin A, meaning cosec A = hypotenuse/opposite side. Essentially, cosec A = 1/sin A, highlighting their inverse relationship.

Sin 30° is calculated using a right-angled triangle with a 30° angle. In such a triangle, the side opposite the 30° angle is half the hypotenuse. Therefore, sin 30° = opposite/hypotenuse = 1/2. This is a standard value often used in trigonometric problems.

The value of sin A is always ≤ 1 because the hypotenuse is the longest side in a right-angled triangle. Since sin A = opposite/hypotenuse, the numerator (opposite side) can never exceed the denominator (hypotenuse), keeping sin A within the range of 0 to 1.

The Pythagorean identity states that sin² A + cos² A = 1 for any angle A. It derives from the Pythagorean theorem applied to a right-angled triangle. This identity is fundamental for simplifying trigonometric expressions and solving equations.

Tan A can be expressed as the ratio of sin A to cos A, i.e., tan A = sin A/cos A. This relationship is useful for converting between trigonometric functions and simplifying expressions. For example, if sin A = 3/5 and cos A = 4/5, then tan A = (3/5)/(4/5) = 3/4.

For 0°, sin 0° = 0, cos 0° = 1, tan 0° = 0, cosec 0° is undefined, sec 0° = 1, and cot 0° is undefined. For 90°, sin 90° = 1, cos 90° = 0, tan 90° is undefined, cosec 90° = 1, sec 90° is undefined, and cot 90° = 0.

Starting from the identity sin² A + cos² A = 1, divide both sides by cos² A to get (sin² A/cos² A) + (cos² A/cos² A) = 1/cos² A. This simplifies to tan² A + 1 = sec² A, and rearranging gives sec² A - tan² A = 1.

Trigonometry is crucial in real-world applications like architecture, navigation, and physics. For instance, it helps architects design buildings by calculating heights and distances, and pilots use it for navigation by determining angles and distances between points.

Cos 45° is found using an isosceles right-angled triangle where the two legs are equal. If each leg is 1 unit, the hypotenuse is √2 units. Thus, cos 45° = adjacent/hypotenuse = 1/√2, which can be rationalized to √2/2.

Cot A is the reciprocal of tan A, meaning cot A = 1/tan A. Alternatively, cot A can also be expressed as cos A/sin A. This relationship is useful for converting between these trigonometric functions in equations and identities.

To solve a right-angled triangle, use trigonometric ratios to find missing sides or angles. For example, if one angle and one side are known, use sin, cos, or tan to find other sides. The Pythagorean theorem can also be used to find the third side if two sides are known.

Tan 60° is √3, derived from an equilateral triangle divided into two 30-60-90 right triangles. In such a triangle, the sides are in the ratio 1:√3:2. Thus, tan 60° = opposite/adjacent = √3/1 = √3.

Cosec A cannot be less than 1 because it is the reciprocal of sin A, and sin A has a maximum value of 1. Since cosec A = 1/sin A, and sin A ≤ 1, cosec A must be ≥ 1. For example, if sin A = 1/2, cosec A = 2.

Using the identity sec² A = 1 + tan² A, tan A = √(sec² A - 1). Then, sin A = tan A/sec A = √(sec² A - 1)/sec A, and cos A = 1/sec A. Cosec A = 1/sin A = sec A/√(sec² A - 1), and cot A = 1/tan A = 1/√(sec² A - 1).

The identity is cot² A + 1 = cosec² A. It is derived from the Pythagorean identity sin² A + cos² A = 1 by dividing both sides by sin² A. This gives 1 + cot² A = cosec² A, which is useful for simplifying trigonometric expressions.

To find the angle when a trigonometric ratio is known, use the inverse trigonometric functions. For example, if sin A = 0.5, then A = sin⁻¹(0.5) = 30°. These functions are available on scientific calculators and are essential for solving trigonometric equations.

A common mnemonic is SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. This helps in recalling the definitions of the primary trigonometric ratios quickly and accurately.

Consider a right-angled triangle with hypotenuse 1. By definition, sin A = opposite side and cos A = adjacent side. According to the Pythagorean theorem, opposite² + adjacent² = hypotenuse², which translates to sin² A + cos² A = 1, verifying the identity.

The angle of elevation is the angle between the horizontal line and the line of sight to an object above the horizontal. It is used in trigonometry to calculate heights and distances, such as determining the height of a tower or a mountain from a certain distance.

Sec 30° is the reciprocal of cos 30°. Since cos 30° = √3/2, sec 30° = 1/cos 30° = 2/√3, which can be rationalized to 2√3/3. This value is often used in trigonometric calculations involving 30° angles.

The angle of elevation is the angle above the horizontal line when looking up at an object, while the angle of depression is the angle below the horizontal line when looking down at an object. Both are used in trigonometry to solve problems involving heights and distances.

To calculate the length of a shadow, use the tangent of the angle of elevation of the sun. For example, if the sun's angle of elevation is 45° and the height of an object is h, the shadow length is h/tan 45° = h, since tan 45° = 1.