Solving Trigonometric Equations: A Step-by-Step Guide
Solving Trigonometric Equations: A Step-by-Step Guide
When faced with a problem involving trigonometric identities, you can use a variety of techniques to simplify and solve for the unknowns. This guide will walk you through solving a specific example involving expressions for sine, cosine, and tangent.
Understanding Trigonometric Identities
In trigonometry, identities are equations that are true for all values of the variables within their domain. One such identity is:
[sec^2theta - tan^2theta 1]
This identity can be used to solve for trigonometric functions like cosine and secant. Let's go through an example step-by-step.
Example Problem
Given that (tantheta x), we can solve for (costheta) using trigonometric identities.
Step 1: Use the Identity
Starting with the identity (sec^2theta - tan^2theta 1), we can substitute (tantheta x). The identity becomes:
[sec^2theta - x^2 1]
From this, we can solve for (sec^2theta):
[sec^2theta x^2 1]
Step 2: Solve for (costheta)
Recall that (sectheta frac{1}{costheta}), so:
[cos^2theta frac{1}{sec^2theta}]
Substituting (sec^2theta x^2 1) into the equation gives:
[cos^2theta frac{1}{x^2 1}]
Therefore:
[costheta pm sqrt{frac{1}{x^2 1}}]
However, since (theta) is in the second quadrant, where the cosine is negative, we choose the negative root:
[costheta -sqrt{frac{1}{x^2 1}}]
Alternative Approach
Another way to solve the problem is by using the Pythagorean identity and the definition of tangent. Here's the detailed step-by-step process:
Step 1: Use the Pythagorean Identity
The Pythagorean identity states:
[sin^2theta cos^2theta 1]
We can also express tangent in terms of sine and cosine:
[tantheta frac{sintheta}{costheta}]
Given that (tantheta x), we can write:
[sintheta x costheta]
Squaring both sides gives:
[sin^2theta x^2 cos^2theta]
Step 2: Substitute into the Pythagorean Identity
Substitute (sin^2theta x^2 cos^2theta) into the Pythagorean identity:
[x^2 cos^2theta cos^2theta 1]
We can factor out (cos^2theta):
[cos^2theta (x^2 1) 1]
Isolating (cos^2theta) gives:
[cos^2theta frac{1}{x^2 1}]
Since (costheta) is negative in the second quadrant:
[costheta -sqrt{frac{1}{x^2 1}}]
Conclusion
By following these steps, you can solve trigonometric equations involving various identities. It's crucial to identify the quadrants and the signs of the trigonometric functions to find the correct solution. Always double-check your work and ensure you understand each step of the process.