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Solving Trigonometric Equations Using the Tangent Function

Solving Trigonometric Equations Using the Tangent Function

January 06, 2025 Education

Solving Trigonometric Equations Using the Tangent Function

Introduction

Trigonometry is a fundamental branch of mathematics that deals with the relationships between the angles and sides of triangles. Understanding trigonometric functions, such as the tangent, is crucial for solving a wide range of mathematical problems. In this article, we will explore how to solve a class 11 trigonometry problem using the tangent function.

Problem Statement

The problem at hand involves finding the values of a, b, and c in the equation y /tancx

Step-by-Step Solution

Identify the Parent Graph

First, we need to recognize that the given function is y /tancx

Based on the problem statement, we need to determine whether the function is tangent or cotangent. The y-value of the function increases from left to right, indicating that it is tangent, not cotangent. For cotangent, the y-value decreases from right to left.

Determine the Y-Intercept

To start solving, let's substitute x 0 into the equation:

y /tan(0) y /0 Simplify: tan(0) 0 y

From the graph, we can see that the y-intercept is 2. Therefore, a 2.

Find the Period

The period of the tangent function can be determined using the formula Period pi/b. From the graph, we observe that the function completes one period at x pi/2.

To find b, we use the equation:

pi/b pi/2 b 2 Solve for c

Next, we need to find the value of c. Substituting the given point (-pi/8, -1) into the equation:

y 2/tan(c * -pi/8) -1 2/tan(-pi/4) tan(-pi/4) -1 (since the tangent of -pi/4 is -1) -1 2/(-1) -1 -2 1 2 c 2

Final Equation

Substituting the values of a, b, and c into the equation, we get:

y 2*3*tan(2x) 6*tan(2x)

This is the final equation that satisfies the given problem.

Additional Methods

For a deeper understanding, let's explore an alternative approach using a special case. Suppose a 30o, b 60o, and c 90o. We will check if the equation satisfies the identity on the right-hand side.

Right-Hand Side (RHS) Analysis

The RHS of the equation is given by:

RHS 4cos(a*b/2)cos(b*c/2)cos(c*a/2)

RHS 4cos(30*60/2)cos(60*90/2)cos(90*30/2) RHS 4cos(900/2)cos(5400/2)cos(2700/2) RHS 4cos(450)cos(2700)cos(1350) RHS 4cos(90o)cos(0o)cos(45o) RHS 4*0*cos(45o) RHS 0

Since the RHS is 0, we see that the special case does not directly satisfy the equation as intended. However, this method helps us understand the properties of the trigonometric functions involved.

Approaching from RHS to LHS

To approach the problem from the RHS to LHS, we will use trigonometric identities to simplify the equation and show that the left-hand side (LHS) equals the RHS:

RHS 4cos(a*b/2)cos(b*c/2)cos(c*a/2) RHS 2[cos(ab/2)cos(bc/2)cos(ca/2)] RHS 2[cos(a(b-c)/2)cos(ca/2)] RHS 2cos(a(b-c)/2)2cos(ca/2)2cos(ca/2) RHS cos(abca/2)cos(a-2bc/2)cos(a-ca/2)cos(a-c-c-a/2) RHS cos(abc)cos(bc)cos(ac)cos(bc) RHS LHS

This confirms that the identity holds true. By using trigonometric identities, we can simplify and verify the given problem.

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