Solving for tan A Given Sin A 1/2: A Comprehensive Guide

Introduction to Trigonometric Functions

Understanding the Problem

When solving trigonometric equations, one common scenario involves finding the value of one trigonometric function given the value of another. This tutorial will guide you through solving for tan A when sin A is given as 1/2. This process involves the use of fundamental trigonometric identities and properties of right-angled triangles. Let's explore this step by step.

Step 1: Determining the Angle A

Given:

sinA12To find the angle A, we need to take the inverse sine (aka arcsin) of both sides:

Asin?112

This gives us:

A30°Now that we have the angle, we can proceed to find tan A.

Step 2: Solving for tan 30° Using Identities

One method to find tan A is by using the identity:

sin2A cos2A1Given sin A 1/2, we can substitute and solve for cos A:

sin2A14cosA21?1434cos A3432Now that we have cos A, we can find tan A using the formula:

tan Asin Acos Atan 30°123213Step 3: Using a Right-Angled Triangle

An alternative method to solve this problem involves using the properties of a right-angled triangle. Let's use a 30-60-90 triangle, where:

The side opposite to 30° is 1/2 the hypotenuse. The side opposite to 60° is √3/2 the hypotenuse. The hypotenuse is 1 (assuming unit length for simplicity).

From this, we can see that:

The side opposite to 30° is 1/2. The side adjacent to 30° is √3/2.

Therefore, using the definition of tangent:

tan 30°oppositeadjacent123213Both methods confirm that:

tan 30°13Conclusion

This tutorial has shown how to use both trigonometric identities and the right-angled triangle method to solve for the tangent of an angle given its sine value. It's important to understand these fundamental concepts to tackle more complex trigonometric problems effectively. Stay tuned for more guides and tutorials in trigonometry and related areas.