Trigonometrical Identities - Class 10 MCQ

If sin A = 0.5, then by the reciprocal identity, cosec A = 1 / sin A = 1 / 0.5 = 2. This means that the cosecant of angle A, which is the reciprocal of the sine function, is 2. Understanding these relationships is crucial for solving trigonometric equations and simplifying expressions.

The sine ratio for an angle A in a right-angled triangle is defined as the ratio of the length of the side opposite the angle (Perpendicular) to the length of the longest side (Hypotenuse). Therefore, the formula for sine is given by sin A = Perpendicular / Hypotenuse. This relationship is crucial in trigonometry as it allows for the calculation of unknown sides and angles in various applications.

Trigonometric tables provide values of sine, cosine, and tangent for various angles, typically listed in degrees and minutes. These tables are essential tools for quickly finding trigonometric values for calculations, especially before the advent of calculators. They allow mathematicians and engineers to perform calculations efficiently and accurately.

The formula sin² A + cos² A = 1 is a fundamental square relation in trigonometry, derived from the Pythagorean theorem. This identity holds true for all angles and is pivotal in establishing relationships between the different trigonometric functions, enabling the simplification and solving of many trigonometric problems.

The relationship between sine and cosine for complementary angles is expressed as sin (90° - A) = cos A. This identity emphasizes the fundamental connection between these two functions, showcasing how they are interrelated when angles add up to 90 degrees. This principle is often applied in various trigonometric calculations.

To solve cos 38° sec (90° - 2A) = 1, we use the identity sec (90° - 2A) = cosec 2A. Thus, cos 38° * (1/sin 2A) = 1 leads to sin 2A = cos 38°. This implies 2A = 52°, so A = 26°. This example highlights the interplay between different trigonometric identities and their applications in solving for angles.

A reciprocal identity defines the relationship between a trigonometric function and its reciprocal. The identity sin A = 1 / cosec A indicates that sine and cosecant are reciprocals of each other. Understanding reciprocal identities is essential for simplifying trigonometric expressions and solving equations.

The tangent of an angle A is defined as the ratio of sine to cosine, expressed mathematically as tan A = sin A / cos A. This relationship allows for the calculation of tangent values based on the sine and cosine functions, which are foundational in trigonometry.

The Pythagorean theorem relates to trigonometric ratios through the identity sin² A + cos² A = 1. This equation illustrates how the squares of the sine and cosine functions of an angle are always equal to 1, reflecting the relationship between the sides of a right triangle. This concept is essential for understanding trigonometric identities and their applications.

The value of tan A can be calculated using the formula tan A = sin A / cos A. Substituting the given values, tan A = (3/5) / (4/5) = 3/4. This illustrates how knowing the sine and cosine values allows for the determination of tangent, which is vital in solving problems involving angles and triangle properties.