Find tan(x) Given sin(x)cos(x) = ⁷ ⁄ ₃₉ : A Step-by-Step Guide

Hey guys! Today, we’re diving into a fun little trigonometry problem where we need to find the value of tan(x) when we know that sin(x)cos(x) = ⁷ ⁄ ₃₉ . Sounds like a mouthful, right? But don’t worry, we’ll break it down step by step so it’s super easy to follow. So, grab your calculators, and let’s get started!

Understanding the Problem

Before we jump into solving, let’s make sure we understand what we’re dealing with. We’re given the value of the product of sin(x) and cos(x), and our mission is to find the value of tan(x). Remember that tan(x) is defined as sin(x) / cos(x). This is a crucial relationship that we’ll use to solve the problem. Also, keep in mind the trigonometric identities, especially the Pythagorean identity, which states that sin^2(x) + cos^2(x) = 1. These identities are the building blocks of trigonometry, and understanding them is key to solving a wide range of problems.

When faced with a problem like this, it’s always a good idea to think about how the given information relates to what we need to find. In this case, we have sin(x)cos(x), and we want to find sin(x) / cos(x). Is there a way to manipulate the given information to get closer to our goal? Absolutely! We can use algebraic manipulations and trigonometric identities to bridge the gap between what we know and what we want to find. For instance, we might try squaring the given equation or using the Pythagorean identity to introduce sin^2(x) and cos^2(x) terms. By carefully considering these strategies, we can develop a plan of attack that leads us to the solution. Trigonometry is all about relationships and how to manipulate them to your advantage. So, let’s put on our thinking caps and get ready to solve this problem!

Step 1: Using Trigonometric Identities

The core idea here is to relate sin(x)cos(x) to tan(x) . We know that tan(x) = sin(x) / cos(x) . To find this, we need to somehow manipulate our given equation. A clever trick is to consider the expression (sin(x) + cos(x))^2 . Let’s expand this:

(sin(x) + cos(x))^2 = sin^2(x) + 2sin(x)cos(x) + cos^2(x)

Now, remember the Pythagorean identity: sin^2(x) + cos^2(x) = 1 . So, we can simplify the above equation to:

(sin(x) + cos(x))^2 = 1 + 2sin(x)cos(x)

We know that sin(x)cos(x) = 7/39 , so let’s substitute that in:

(sin(x) + cos(x))^2 = 1 + 2*(7/39) = 1 + 14/39 = 53/39

Therefore:

sin(x) + cos(x) = ±√(53/39)

Similarly, let’s look at (sin(x) - cos(x))^2 :

(sin(x) - cos(x))^2 = sin^2(x) - 2sin(x)cos(x) + cos^2(x)

Using the Pythagorean identity again:

(sin(x) - cos(x))^2 = 1 - 2sin(x)cos(x)

Substitute sin(x)cos(x) = 7/39 :

(sin(x) - cos(x))^2 = 1 - 2*(7/39) = 1 - 14/39 = 25/39

So:

sin(x) - cos(x) = ±√(25/39) = ±5/√39

Step 2: Solving for sin(x) and cos(x)

Now we have two equations:

1. sin(x) + cos(x) = ±√(53/39)
2. sin(x) - cos(x) = ±5/√39

We essentially have a system of equations. Let’s solve for sin(x) and cos(x) by considering the different sign combinations.

Case 1: sin(x) + cos(x) = √(53/39) and sin(x) - cos(x) = 5/√39

Adding the two equations:

2sin(x) = √(53/39) + 5/√39

sin(x) = (√(53) + 5) / (2√39)

Subtracting the second equation from the first:

2cos(x) = √(53/39) - 5/√39

cos(x) = (√(53) - 5) / (2√39)

Case 2: sin(x) + cos(x) = √(53/39) and sin(x) - cos(x) = -5/√39

Adding the two equations:

2sin(x) = √(53/39) - 5/√39

sin(x) = (√(53) - 5) / (2√39)

Subtracting the second equation from the first:

2cos(x) = √(53/39) + 5/√39

cos(x) = (√(53) + 5) / (2√39)

Case 3: sin(x) + cos(x) = -√(53/39) and sin(x) - cos(x) = 5/√39

Adding the two equations:

2sin(x) = -√(53/39) + 5/√39

sin(x) = (-√(53) + 5) / (2√39)

Subtracting the second equation from the first:

2cos(x) = -√(53/39) - 5/√39

cos(x) = (-√(53) - 5) / (2√39)

Case 4: sin(x) + cos(x) = -√(53/39) and sin(x) - cos(x) = -5/√39

Adding the two equations:

2sin(x) = -√(53/39) - 5/√39

sin(x) = (-√(53) - 5) / (2√39)

Subtracting the second equation from the first:

2cos(x) = -√(53/39) + 5/√39

cos(x) = (-√(53) + 5) / (2√39)

Step 3: Calculate tan(x)

Now that we have sin(x) and cos(x) for each case, we can calculate tan(x) = sin(x) / cos(x) .

Case 1:

tan(x) = ((√(53) + 5) / (2√39)) / ((√(53) - 5) / (2√39)) = (√(53) + 5) / (√(53) - 5)

To rationalize the denominator, multiply by (√(53) + 5) / (√(53) + 5) :

tan(x) = (√(53) + 5)^2 / (53 - 25) = (53 + 10√(53) + 25) / 28 = (78 + 10√(53)) / 28 = (39 + 5√(53)) / 14

Case 2:

tan(x) = ((√(53) - 5) / (2√39)) / ((√(53) + 5) / (2√39)) = (√(53) - 5) / (√(53) + 5)

To rationalize the denominator, multiply by (√(53) - 5) / (√(53) - 5) :

tan(x) = (√(53) - 5)^2 / (53 - 25) = (53 - 10√(53) + 25) / 28 = (78 - 10√(53)) / 28 = (39 - 5√(53)) / 14

Case 3:

tan(x) = ((-√(53) + 5) / (2√39)) / ((-√(53) - 5) / (2√39)) = (-√(53) + 5) / (-√(53) - 5) = (5 - √(53)) / (-5 - √(53))

Multiply by (-5 + √(53)) / (-5 + √(53)) = (5 - √(53)) / (-5 - √(53)) * (-5 + √(53))/(-5 + √(53)) = (39 - 5√(53)) / 14

Case 4:

tan(x) = ((-√(53) - 5) / (2√39)) / ((-√(53) + 5) / (2√39)) = (-√(53) - 5) / (-√(53) + 5) = (-5 - √(53)) / (5 - √(53))

Multiply by (5 + √(53)) / (5 + √(53)) = (39 + 5√(53)) / 14

Final Answer

So, the possible values for tan(x) are:

• (39 + 5√(53)) / 14
• (39 - 5√(53)) / 14

These are the two possible solutions for tan(x) given the initial condition. Remember to always double-check your work and ensure the solutions make sense within the context of the problem. Great job, guys! You’ve successfully navigated through this trigonometry challenge!