Subjective Questions
Trigonometry and Trigonometric Identities
Chapter 1: Introduction to Trigonometry and Trigonometric Identities
Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It has a wide range of applications in various fields such as physics, engineering, and astronomy. In this chapter, we will explore the fundamental concepts of trigonometry and learn about trigonometric identities, which are essential tools in solving trigonometric equations.
Section 1: Trigonometric Ratios
In this section, we will introduce the basic trigonometric ratios – sine, cosine, and tangent. These ratios are defined in terms of the sides of a right triangle. The sine of an angle is equal to the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine of an angle is equal to the ratio of the length of the adjacent side to the length of the hypotenuse. The tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side. We will also discuss the reciprocal trigonometric ratios – cosecant, secant, and cotangent.
Example 1 (Simple): Find the value of sine of angle A in a right triangle ABC, where the length of the side opposite angle A is 5 cm and the length of the hypotenuse is 13 cm.
Solution: The sine of angle A is equal to the ratio of the length of the side opposite angle A to the length of the hypotenuse. Therefore, sin A = 5/13.
Example 2 (Medium): Find the value of cosine of angle B in a right triangle XYZ, where the length of the adjacent side to angle B is 12 cm and the length of the hypotenuse is 15 cm.
Solution: The cosine of angle B is equal to the ratio of the length of the adjacent side to the length of the hypotenuse. Therefore, cos B = 12/15 = 4/5.
Example 3 (Complex): Find the value of tangent of angle C in a right triangle PQR, where the length of the side opposite angle C is 7 cm and the length of the adjacent side is 24 cm.
Solution: The tangent of angle C is equal to the ratio of the length of the side opposite angle C to the length of the adjacent side. Therefore, tan C = 7/24.
Section 2: Trigonometric Identities
In this section, we will explore trigonometric identities, which are equations involving trigonometric functions that are true for all values of the variables. These identities are useful in simplifying trigonometric expressions and solving trigonometric equations. Some of the important trigonometric identities include the Pythagorean identities, reciprocal identities, quotient identities, and co-function identities.
Example 4 (Simple): Simplify the expression sin^2(x) + cos^2(x).
Solution: According to the Pythagorean identity, sin^2(x) + cos^2(x) = 1.
Example 5 (Medium): Simplify the expression tan(x) + cot(x).
Solution: According to the quotient identity, tan(x) + cot(x) = (sin(x)/cos(x)) + (cos(x)/sin(x)) = (sin^2(x) + cos^2(x))/(sin(x)cos(x)) = 1/(sin(x)cos(x)).
Example 6 (Complex): Simplify the expression sec^2(x) – tan^2(x).
Solution: According to the Pythagorean identity, sec^2(x) – tan^2(x) = 1.
Section 3: Trigonometric Equations
In this section, we will learn how to solve trigonometric equations, which are equations involving trigonometric functions. We will use the trigonometric identities and basic algebraic techniques to solve these equations. Some of the common types of trigonometric equations include linear equations, quadratic equations, and equations involving multiple trigonometric functions.
Example 7 (Simple): Solve the equation sin(x) = 1/2.
Solution: We can solve this equation by taking the inverse sine (or arcsine) of both sides. Therefore, x = arcsin(1/2) = π/6 or 5π/6.
Example 8 (Medium): Solve the equation cos(2x) = 0.
Solution: We can solve this equation by using the double angle identity for cosine. Therefore, 2x = π/2 + kπ or 2x = 3π/2 + kπ, where k is an integer. Simplifying, we get x = π/4 + kπ/2 or x = 3π/4 + kπ/2, where k is an integer.
Example 9 (Complex): Solve the equation tan^2(x) + 2tan(x) – 1 = 0.
Solution: We can solve this equation by using the quadratic formula. Let tan(x) = t. Therefore, t^2 + 2t – 1 = 0. Solving this quadratic equation, we get t = (-2 ± √(4 + 4))/2 = (-2 ± √8)/2 = -1 ± √2. Substituting back, we get tan(x) = -1 ± √2. Taking the inverse tangent (or arctangent) of both sides, we get x = arctan(-1 + √2) or x = arctan(-1 – √2).
In this chapter, we have covered the fundamental concepts of trigonometry and trigonometric identities. We have learned about trigonometric ratios, trigonometric identities, and how to solve trigonometric equations. These concepts are crucial for understanding more advanced topics in trigonometry and their applications in various fields.